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
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Real exponential function · qf-mod44-exp-zero-inverse-positivity · candidate
Canonical question JSONL · English JSONL · Validation report