# Exact Finite Sums, Whole Exponential Identities and Convolution Scope

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batch-20261015-advance-20261113-r02-real-exponential-function-foundations · 2026-10-15 · 100 questions

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## 1. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=1/2, cutoff=5, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0001/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 2. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=2/3, cutoff=6, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0002/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 3. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=3/4, cutoff=7, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0003/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 4. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=4/5, cutoff=8, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0004/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 5. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=5/6, cutoff=9, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0005/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 6. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=6/7, cutoff=10, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0006/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 7. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=7/8, cutoff=11, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0007/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 8. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=8/9, cutoff=12, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0008/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 9. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=9/10, cutoff=13, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000], 630288054851920098713/256256000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000], 630288054851920098713/256256000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0009/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 10. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=10/11, cutoff=14, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0010/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 11. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=11/12, cutoff=15, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0011/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 12. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=12/13, cutoff=16, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0012/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 13. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=13/14, cutoff=17, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0013/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 14. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=14/15, cutoff=18, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0014/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 15. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=15/16, cutoff=19, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0015/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 16. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=16/17, cutoff=20, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0016/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 17. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=17/18, cutoff=21, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0017/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 18. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=18/19, cutoff=22, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0018/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 19. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=19/20, cutoff=23, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0019/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 20. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=20/21, cutoff=24, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0020/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 21. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=21/22, cutoff=25, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0021/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 22. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=22/23, cutoff=26, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0022/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 23. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=23/24, cutoff=27, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0023/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 24. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=24/25, cutoff=28, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0024/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 25. Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=25/26, cutoff=29, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary).

The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0.

View answer and explanation(25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, No, finite_partial_sum_is_not_the_infinite_exponential_value)

Compute every exact rational term and cumulative partial sum with factorial arithmetic. The fixed No withholds a general infinite-value certificate, not an inequality at every x; at x=0 a finite sum may already equal E(0)=1. No approximate rational value is promoted to the exact exponential at a nonzero input. The exact result, retaining the requested order, is (25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, No, finite_partial_sum_is_not_the_infinite_exponential_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0025/)

Real exponential function · qf-mod44-exp-finite-partial-sum · candidate





## 26. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/2, N=2, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0026/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 27. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/3, N=4, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0027/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 28. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/4, N=6, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0028/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 29. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/5, N=5, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0029/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 30. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/6, N=7, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0030/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 31. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/7, N=9, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0031/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 32. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/8, N=8, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0032/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 33. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/9, N=10, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0033/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 34. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/10, N=12, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0034/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 35. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/11, N=11, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0035/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 36. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/12, N=13, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0036/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 37. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/13, N=15, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0037/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 38. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/14, N=14, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0038/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 39. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/15, N=16, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0039/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 40. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/16, N=18, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0040/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 41. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/17, N=17, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0041/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 42. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/18, N=19, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0042/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 43. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/19, N=21, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0043/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 44. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/20, N=20, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0044/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 45. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/21, N=22, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0045/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 46. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/22, N=24, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0046/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 47. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/23, N=23, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0047/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 48. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/24, N=25, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0048/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 49. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/25, N=27, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0049/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 50. The Whole Factorial Coefficient Ratio Supplies an All-Later Witness Recipe

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use epsilon=1/26, N=26, requiring N at least the denominator. Give (epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate).

This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff.

View answer and explanation(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

For coefficients a_n=1/n!, |a_(n+1)/a_n|=1/(n+1) and every n≥N gives a value at most1/(N+1)<epsilon. The full symbolic recipe supplies such N for arbitrary positive epsilon, not only one cutoff observation. Terms after substituting a fixed x have the extra |x| factor and still ratio limit0, so the radius is infinity. The exact result, retaining the requested order, is (26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, Yes, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0050/)

Real exponential function · qf-mod44-exp-infinite-radius-certificate · candidate





## 51. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=1/2, y=-2/3, declared degree=0. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(1/2, -2/3, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (1/2, -2/3, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0051/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 52. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=2/3, y=-3/4, declared degree=1. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0052/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 53. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=3/4, y=-4/5, declared degree=2. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0053/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 54. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=4/5, y=-5/6, declared degree=3. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0054/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 55. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=5/6, y=-6/7, declared degree=4. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0055/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 56. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=6/7, y=-7/8, declared degree=5. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0056/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 57. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=7/8, y=-8/9, declared degree=6. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0057/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 58. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=8/9, y=-9/10, declared degree=7. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0058/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 59. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=9/10, y=-10/11, declared degree=8. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0059/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 60. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=10/11, y=-11/12, declared degree=9. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0060/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 61. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=11/12, y=-12/13, declared degree=10. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0061/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 62. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=12/13, y=-13/14, declared degree=11. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0062/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 63. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=13/14, y=-14/15, declared degree=12. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0063/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 64. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=14/15, y=-15/16, declared degree=13. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0064/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 65. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=15/16, y=-16/17, declared degree=14. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0065/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 66. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=16/17, y=-17/18, declared degree=15. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0066/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 67. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=17/18, y=-18/19, declared degree=16. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0067/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 68. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=18/19, y=-19/20, declared degree=17. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0068/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 69. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=19/20, y=-20/21, declared degree=18. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0069/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 70. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=20/21, y=-21/22, declared degree=19. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0070/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 71. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=21/22, y=-22/23, declared degree=20. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0071/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 72. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=22/23, y=-23/24, declared degree=0. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(22/23, -23/24, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (22/23, -23/24, 0, [[0, 1]], 1, 1, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0072/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 73. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=23/24, y=-24/25, declared degree=1. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0073/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 74. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=24/25, y=-25/26, declared degree=2. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0074/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 75. One Convolution Degree Is Not the Whole Exponential Addition Proof

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use x=25/26, y=-26/27, declared degree=3. Give (x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate).

View answer and explanation(25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)

Sum [k,x^k*y^(degree−k)/(k!*(degree−k)!)] over every k=0..degree and compare with (x+y)^degree/degree!. This checks one coefficient. The general addition identity needs the binomial relation at every degree and absolute convergence to rearrange the Cauchy product. The exact result, retaining the requested order, is (25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, Yes, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0075/)

Real exponential function · qf-mod44-exp-addition-convolution · candidate





## 76. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-12. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0076/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 77. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-11. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0077/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 78. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-10. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0078/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 79. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-9. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0079/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 80. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0080/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 81. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-7. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0081/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 82. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-6. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0082/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 83. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-5. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0083/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 84. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-4. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0084/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 85. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-3. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0085/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 86. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-2. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0086/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 87. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=-1. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0087/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 88. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=0. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0088/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 89. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=1. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0089/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 90. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=2. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0090/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 91. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=3. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0091/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 92. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=4. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0092/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 93. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=5. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0093/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 94. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=6. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0094/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 95. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=7. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0095/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 96. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=8. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0096/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 97. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=9. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0097/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 98. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=10. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0098/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 99. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=11. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0099/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate





## 100. Inverse and Half-Value Nonvanishing Precede Strict Square Positivity

Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use real argument x=12. Give (x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate).

View answer and explanation(12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)

Keep E(0)=1 and the symbolic inverse identities. E(x/2)E(−x/2)=1 makes the half-value nonzero; then E(x)=E(x/2)^2>0 is strict. These are whole-series consequences, not a rational evaluation or an automatic derivative/log/ODE/injectivity claim. The exact result, retaining the requested order, is (12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, Yes, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r02-real-exponential-function-0100/)

Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate



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