Question 24237: Exact Finite Exponential Sums Are Not General Infinite-Value Certificates

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-real-exponential-function

Step-by-step derivation

source roles = [exp-finite-partial-sum]

derived = [[x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary]]

source roles = [0]

derived = [1]

(12/13)^0

Exp ORIGINAL x power0 = 1

Exp term0 power over factorial := div(1, 1)

Exp term0 power over factorial = 1

Exp cumulative sum through0 := sum(0, 1)

Exp cumulative sum through0 = 1

Exp term1 factorial step1 := mul(1, 1)

Exp term1 factorial step1 = 1

(12/13)^1

Exp ORIGINAL x power1 = 12/13

Exp term1 power over factorial := div(12/13, 1)

Exp term1 power over factorial = 12/13

Exp cumulative sum through1 := sum(1, 12/13)

Exp cumulative sum through1 = 25/13

Exp term2 factorial step1 := mul(1, 1)

Exp term2 factorial step1 = 1

Exp term2 factorial step2 := mul(1, 2)

Exp term2 factorial step2 = 2

(12/13)^2

Exp ORIGINAL x power2 = 144/169

Exp term2 power over factorial := div(144/169, 2)

Exp term2 power over factorial = 72/169

Exp cumulative sum through2 := sum(25/13, 72/169)

Exp cumulative sum through2 = 397/169

Exp term3 factorial step1 := mul(1, 1)

Exp term3 factorial step1 = 1

Exp term3 factorial step2 := mul(1, 2)

Exp term3 factorial step2 = 2

Exp term3 factorial step3 := mul(2, 3)

Exp term3 factorial step3 = 6

(12/13)^3

Exp ORIGINAL x power3 = 1728/2197

Exp term3 power over factorial := div(1728/2197, 6)

Exp term3 power over factorial = 288/2197

Exp cumulative sum through3 := sum(397/169, 288/2197)

Exp cumulative sum through3 = 5449/2197

Exp term4 factorial step1 := mul(1, 1)

Exp term4 factorial step1 = 1

Exp term4 factorial step2 := mul(1, 2)

Exp term4 factorial step2 = 2

Exp term4 factorial step3 := mul(2, 3)

Exp term4 factorial step3 = 6

Exp term4 factorial step4 := mul(6, 4)

Exp term4 factorial step4 = 24

(12/13)^4

Exp ORIGINAL x power4 = 20736/28561

Exp term4 power over factorial := div(20736/28561, 24)

Exp term4 power over factorial = 864/28561

Exp cumulative sum through4 := sum(5449/2197, 864/28561)

Exp cumulative sum through4 = 71701/28561

Exp term5 factorial step1 := mul(1, 1)

Exp term5 factorial step1 = 1

Exp term5 factorial step2 := mul(1, 2)

Exp term5 factorial step2 = 2

Exp term5 factorial step3 := mul(2, 3)

Exp term5 factorial step3 = 6

Exp term5 factorial step4 := mul(6, 4)

Exp term5 factorial step4 = 24

Exp term5 factorial step5 := mul(24, 5)

Exp term5 factorial step5 = 120

(12/13)^5

Exp ORIGINAL x power5 = 248832/371293

Exp term5 power over factorial := div(248832/371293, 120)

Exp term5 power over factorial = 10368/1856465

Exp cumulative sum through5 := sum(71701/28561, 10368/1856465)

Exp cumulative sum through5 = 4670933/1856465

Exp term6 factorial step1 := mul(1, 1)

Exp term6 factorial step1 = 1

Exp term6 factorial step2 := mul(1, 2)

Exp term6 factorial step2 = 2

Exp term6 factorial step3 := mul(2, 3)

Exp term6 factorial step3 = 6

Exp term6 factorial step4 := mul(6, 4)

Exp term6 factorial step4 = 24

Exp term6 factorial step5 := mul(24, 5)

Exp term6 factorial step5 = 120

Exp term6 factorial step6 := mul(120, 6)

Exp term6 factorial step6 = 720

(12/13)^6

Exp ORIGINAL x power6 = 2985984/4826809

Exp term6 power over factorial := div(2985984/4826809, 720)

Exp term6 power over factorial = 20736/24134045

Exp cumulative sum through6 := sum(4670933/1856465, 20736/24134045)

Exp cumulative sum through6 = 12148573/4826809

Exp term7 factorial step1 := mul(1, 1)

Exp term7 factorial step1 = 1

Exp term7 factorial step2 := mul(1, 2)

Exp term7 factorial step2 = 2

Exp term7 factorial step3 := mul(2, 3)

Exp term7 factorial step3 = 6

Exp term7 factorial step4 := mul(6, 4)

Exp term7 factorial step4 = 24

Exp term7 factorial step5 := mul(24, 5)

Exp term7 factorial step5 = 120

Exp term7 factorial step6 := mul(120, 6)

Exp term7 factorial step6 = 720

Exp term7 factorial step7 := mul(720, 7)

Exp term7 factorial step7 = 5040

(12/13)^7

Exp ORIGINAL x power7 = 35831808/62748517

Exp term7 power over factorial := div(35831808/62748517, 5040)

Exp term7 power over factorial = 248832/2196198095

Exp cumulative sum through7 := sum(12148573/4826809, 248832/2196198095)

Exp cumulative sum through7 = 5527849547/2196198095

Exp term8 factorial step1 := mul(1, 1)

Exp term8 factorial step1 = 1

Exp term8 factorial step2 := mul(1, 2)

Exp term8 factorial step2 = 2

Exp term8 factorial step3 := mul(2, 3)

Exp term8 factorial step3 = 6

Exp term8 factorial step4 := mul(6, 4)

Exp term8 factorial step4 = 24

Exp term8 factorial step5 := mul(24, 5)

Exp term8 factorial step5 = 120

Exp term8 factorial step6 := mul(120, 6)

Exp term8 factorial step6 = 720

Exp term8 factorial step7 := mul(720, 7)

Exp term8 factorial step7 = 5040

Exp term8 factorial step8 := mul(5040, 8)

Exp term8 factorial step8 = 40320

(12/13)^8

Exp ORIGINAL x power8 = 429981696/815730721

Exp term8 power over factorial := div(429981696/815730721, 40320)

Exp term8 power over factorial = 373248/28550575235

Exp cumulative sum through8 := sum(5527849547/2196198095, 373248/28550575235)

Exp cumulative sum through8 = 71862417359/28550575235

Exp term9 factorial step1 := mul(1, 1)

Exp term9 factorial step1 = 1

Exp term9 factorial step2 := mul(1, 2)

Exp term9 factorial step2 = 2

Exp term9 factorial step3 := mul(2, 3)

Exp term9 factorial step3 = 6

Exp term9 factorial step4 := mul(6, 4)

Exp term9 factorial step4 = 24

Exp term9 factorial step5 := mul(24, 5)

Exp term9 factorial step5 = 120

Exp term9 factorial step6 := mul(120, 6)

Exp term9 factorial step6 = 720

Exp term9 factorial step7 := mul(720, 7)

Exp term9 factorial step7 = 5040

Exp term9 factorial step8 := mul(5040, 8)

Exp term9 factorial step8 = 40320

Exp term9 factorial step9 := mul(40320, 9)

Exp term9 factorial step9 = 362880

(12/13)^9

Exp ORIGINAL x power9 = 5159780352/10604499373

Exp term9 power over factorial := div(5159780352/10604499373, 362880)

Exp term9 power over factorial = 497664/371157478055

Exp cumulative sum through9 := sum(71862417359/28550575235, 497664/371157478055)

Exp cumulative sum through9 = 934211923331/371157478055

Exp term10 factorial step1 := mul(1, 1)

Exp term10 factorial step1 = 1

Exp term10 factorial step2 := mul(1, 2)

Exp term10 factorial step2 = 2

Exp term10 factorial step3 := mul(2, 3)

Exp term10 factorial step3 = 6

Exp term10 factorial step4 := mul(6, 4)

Exp term10 factorial step4 = 24

Exp term10 factorial step5 := mul(24, 5)

Exp term10 factorial step5 = 120

Exp term10 factorial step6 := mul(120, 6)

Exp term10 factorial step6 = 720

Exp term10 factorial step7 := mul(720, 7)

Exp term10 factorial step7 = 5040

Exp term10 factorial step8 := mul(5040, 8)

Exp term10 factorial step8 = 40320

Exp term10 factorial step9 := mul(40320, 9)

Exp term10 factorial step9 = 362880

Exp term10 factorial step10 := mul(362880, 10)

Exp term10 factorial step10 = 3628800

(12/13)^10

Exp ORIGINAL x power10 = 61917364224/137858491849

Exp term10 power over factorial := div(61917364224/137858491849, 3628800)

Exp term10 power over factorial = 2985984/24125236073575

Exp cumulative sum through10 := sum(934211923331/371157478055, 2985984/24125236073575)

Exp cumulative sum through10 = 60723778002499/24125236073575

Exp term11 factorial step1 := mul(1, 1)

Exp term11 factorial step1 = 1

Exp term11 factorial step2 := mul(1, 2)

Exp term11 factorial step2 = 2

Exp term11 factorial step3 := mul(2, 3)

Exp term11 factorial step3 = 6

Exp term11 factorial step4 := mul(6, 4)

Exp term11 factorial step4 = 24

Exp term11 factorial step5 := mul(24, 5)

Exp term11 factorial step5 = 120

Exp term11 factorial step6 := mul(120, 6)

Exp term11 factorial step6 = 720

Exp term11 factorial step7 := mul(720, 7)

Exp term11 factorial step7 = 5040

Exp term11 factorial step8 := mul(5040, 8)

Exp term11 factorial step8 = 40320

Exp term11 factorial step9 := mul(40320, 9)

Exp term11 factorial step9 = 362880

Exp term11 factorial step10 := mul(362880, 10)

Exp term11 factorial step10 = 3628800

Exp term11 factorial step11 := mul(3628800, 11)

Exp term11 factorial step11 = 39916800

(12/13)^11

Exp ORIGINAL x power11 = 743008370688/1792160394037

Exp term11 power over factorial := div(743008370688/1792160394037, 39916800)

Exp term11 power over factorial = 35831808/3449908758521225

Exp cumulative sum through11 := sum(60723778002499/24125236073575, 35831808/3449908758521225)

Exp cumulative sum through11 = 248100008291119/98568821672035

Exp term12 factorial step1 := mul(1, 1)

Exp term12 factorial step1 = 1

Exp term12 factorial step2 := mul(1, 2)

Exp term12 factorial step2 = 2

Exp term12 factorial step3 := mul(2, 3)

Exp term12 factorial step3 = 6

Exp term12 factorial step4 := mul(6, 4)

Exp term12 factorial step4 = 24

Exp term12 factorial step5 := mul(24, 5)

Exp term12 factorial step5 = 120

Exp term12 factorial step6 := mul(120, 6)

Exp term12 factorial step6 = 720

Exp term12 factorial step7 := mul(720, 7)

Exp term12 factorial step7 = 5040

Exp term12 factorial step8 := mul(5040, 8)

Exp term12 factorial step8 = 40320

Exp term12 factorial step9 := mul(40320, 9)

Exp term12 factorial step9 = 362880

Exp term12 factorial step10 := mul(362880, 10)

Exp term12 factorial step10 = 3628800

Exp term12 factorial step11 := mul(3628800, 11)

Exp term12 factorial step11 = 39916800

Exp term12 factorial step12 := mul(39916800, 12)

Exp term12 factorial step12 = 479001600

(12/13)^12

Exp ORIGINAL x power12 = 8916100448256/23298085122481

Exp term12 power over factorial := div(8916100448256/23298085122481, 479001600)

Exp term12 power over factorial = 35831808/44848813860775925

Exp cumulative sum through12 := sum(248100008291119/98568821672035, 35831808/44848813860775925)

Exp cumulative sum through12 = 112885503808290953/44848813860775925

Exp term13 factorial step1 := mul(1, 1)

Exp term13 factorial step1 = 1

Exp term13 factorial step2 := mul(1, 2)

Exp term13 factorial step2 = 2

Exp term13 factorial step3 := mul(2, 3)

Exp term13 factorial step3 = 6

Exp term13 factorial step4 := mul(6, 4)

Exp term13 factorial step4 = 24

Exp term13 factorial step5 := mul(24, 5)

Exp term13 factorial step5 = 120

Exp term13 factorial step6 := mul(120, 6)

Exp term13 factorial step6 = 720

Exp term13 factorial step7 := mul(720, 7)

Exp term13 factorial step7 = 5040

Exp term13 factorial step8 := mul(5040, 8)

Exp term13 factorial step8 = 40320

Exp term13 factorial step9 := mul(40320, 9)

Exp term13 factorial step9 = 362880

Exp term13 factorial step10 := mul(362880, 10)

Exp term13 factorial step10 = 3628800

Exp term13 factorial step11 := mul(3628800, 11)

Exp term13 factorial step11 = 39916800

Exp term13 factorial step12 := mul(39916800, 12)

Exp term13 factorial step12 = 479001600

Exp term13 factorial step13 := mul(479001600, 13)

Exp term13 factorial step13 = 6227020800

(12/13)^13

Exp ORIGINAL x power13 = 106993205379072/302875106592253

Exp term13 power over factorial := div(106993205379072/302875106592253, 6227020800)

Exp term13 power over factorial = 429981696/7579449542471131325

Exp cumulative sum through13 := sum(112885503808290953/44848813860775925, 429981696/7579449542471131325)

Exp cumulative sum through13 = 19077650144031152753/7579449542471131325

Exp term14 factorial step1 := mul(1, 1)

Exp term14 factorial step1 = 1

Exp term14 factorial step2 := mul(1, 2)

Exp term14 factorial step2 = 2

Exp term14 factorial step3 := mul(2, 3)

Exp term14 factorial step3 = 6

Exp term14 factorial step4 := mul(6, 4)

Exp term14 factorial step4 = 24

Exp term14 factorial step5 := mul(24, 5)

Exp term14 factorial step5 = 120

Exp term14 factorial step6 := mul(120, 6)

Exp term14 factorial step6 = 720

Exp term14 factorial step7 := mul(720, 7)

Exp term14 factorial step7 = 5040

Exp term14 factorial step8 := mul(5040, 8)

Exp term14 factorial step8 = 40320

Exp term14 factorial step9 := mul(40320, 9)

Exp term14 factorial step9 = 362880

Exp term14 factorial step10 := mul(362880, 10)

Exp term14 factorial step10 = 3628800

Exp term14 factorial step11 := mul(3628800, 11)

Exp term14 factorial step11 = 39916800

Exp term14 factorial step12 := mul(39916800, 12)

Exp term14 factorial step12 = 479001600

Exp term14 factorial step13 := mul(479001600, 13)

Exp term14 factorial step13 = 6227020800

Exp term14 factorial step14 := mul(6227020800, 14)

Exp term14 factorial step14 = 87178291200

(12/13)^14

Exp ORIGINAL x power14 = 1283918464548864/3937376385699289

Exp term14 power over factorial := div(1283918464548864/3937376385699289, 87178291200)

Exp term14 power over factorial = 2579890176/689729908364872950575

Exp cumulative sum through14 := sum(19077650144031152753/7579449542471131325, 2579890176/689729908364872950575)

Exp cumulative sum through14 = 1736066163109414790699/689729908364872950575

Exp term15 factorial step1 := mul(1, 1)

Exp term15 factorial step1 = 1

Exp term15 factorial step2 := mul(1, 2)

Exp term15 factorial step2 = 2

Exp term15 factorial step3 := mul(2, 3)

Exp term15 factorial step3 = 6

Exp term15 factorial step4 := mul(6, 4)

Exp term15 factorial step4 = 24

Exp term15 factorial step5 := mul(24, 5)

Exp term15 factorial step5 = 120

Exp term15 factorial step6 := mul(120, 6)

Exp term15 factorial step6 = 720

Exp term15 factorial step7 := mul(720, 7)

Exp term15 factorial step7 = 5040

Exp term15 factorial step8 := mul(5040, 8)

Exp term15 factorial step8 = 40320

Exp term15 factorial step9 := mul(40320, 9)

Exp term15 factorial step9 = 362880

Exp term15 factorial step10 := mul(362880, 10)

Exp term15 factorial step10 = 3628800

Exp term15 factorial step11 := mul(3628800, 11)

Exp term15 factorial step11 = 39916800

Exp term15 factorial step12 := mul(39916800, 12)

Exp term15 factorial step12 = 479001600

Exp term15 factorial step13 := mul(479001600, 13)

Exp term15 factorial step13 = 6227020800

Exp term15 factorial step14 := mul(6227020800, 14)

Exp term15 factorial step14 = 87178291200

Exp term15 factorial step15 := mul(87178291200, 15)

Exp term15 factorial step15 = 1307674368000

(12/13)^15

Exp ORIGINAL x power15 = 15407021574586368/51185893014090757

Exp term15 power over factorial := div(15407021574586368/51185893014090757, 1307674368000)

Exp term15 power over factorial = 10319560704/44832444043716741787375

Exp cumulative sum through15 := sum(1736066163109414790699/689729908364872950575, 10319560704/44832444043716741787375)

Exp cumulative sum through15 = 112844300602122280956139/44832444043716741787375

Exp term16 factorial step1 := mul(1, 1)

Exp term16 factorial step1 = 1

Exp term16 factorial step2 := mul(1, 2)

Exp term16 factorial step2 = 2

Exp term16 factorial step3 := mul(2, 3)

Exp term16 factorial step3 = 6

Exp term16 factorial step4 := mul(6, 4)

Exp term16 factorial step4 = 24

Exp term16 factorial step5 := mul(24, 5)

Exp term16 factorial step5 = 120

Exp term16 factorial step6 := mul(120, 6)

Exp term16 factorial step6 = 720

Exp term16 factorial step7 := mul(720, 7)

Exp term16 factorial step7 = 5040

Exp term16 factorial step8 := mul(5040, 8)

Exp term16 factorial step8 = 40320

Exp term16 factorial step9 := mul(40320, 9)

Exp term16 factorial step9 = 362880

Exp term16 factorial step10 := mul(362880, 10)

Exp term16 factorial step10 = 3628800

Exp term16 factorial step11 := mul(3628800, 11)

Exp term16 factorial step11 = 39916800

Exp term16 factorial step12 := mul(39916800, 12)

Exp term16 factorial step12 = 479001600

Exp term16 factorial step13 := mul(479001600, 13)

Exp term16 factorial step13 = 6227020800

Exp term16 factorial step14 := mul(6227020800, 14)

Exp term16 factorial step14 = 87178291200

Exp term16 factorial step15 := mul(87178291200, 15)

Exp term16 factorial step15 = 1307674368000

Exp term16 factorial step16 := mul(1307674368000, 16)

Exp term16 factorial step16 = 20922789888000

(12/13)^16

Exp ORIGINAL x power16 = 184884258895036416/665416609183179841

Exp term16 power over factorial := div(184884258895036416/665416609183179841, 20922789888000)

Exp term16 power over factorial = 7739670528/582821772568317643235875

Exp cumulative sum through16 := sum(112844300602122280956139/44832444043716741787375, 7739670528/582821772568317643235875)

Exp cumulative sum through16 = 293395181565519478420067/116564354513663528647175

source roles = [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]]

derived = [293395181565519478420067/116564354513663528647175, No, finite sum is not a general infinite-value certificate]

Reverse checks

• S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S189 · premise_and_result_audit — Recompute Exp cumulative sum through16 with exact rational arithmetic, not rounded decimals.
• S188 · premise_and_result_audit — Recompute Exp term16 power over factorial with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S186 · premise_and_result_audit — Recompute Exp term16 factorial step16 with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Recompute Exp term16 factorial step15 with exact rational arithmetic, not rounded decimals.
• S184 · premise_and_result_audit — Recompute Exp term16 factorial step14 with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Recompute Exp term16 factorial step13 with exact rational arithmetic, not rounded decimals.
• S182 · premise_and_result_audit — Recompute Exp term16 factorial step12 with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Exp term16 factorial step11 with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Exp term16 factorial step10 with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Exp term16 factorial step9 with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Recompute Exp term16 factorial step8 with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Recompute Exp term16 factorial step7 with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Exp term16 factorial step6 with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Exp term16 factorial step5 with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute Exp term16 factorial step4 with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Recompute Exp term16 factorial step3 with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Recompute Exp term16 factorial step2 with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute Exp term16 factorial step1 with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Recompute Exp cumulative sum through15 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute Exp term15 power over factorial with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Recompute Exp term15 factorial step15 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute Exp term15 factorial step14 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Recompute Exp term15 factorial step13 with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute Exp term15 factorial step12 with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Recompute Exp term15 factorial step11 with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Recompute Exp term15 factorial step10 with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Exp term15 factorial step9 with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Exp term15 factorial step8 with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Exp term15 factorial step7 with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Recompute Exp term15 factorial step6 with exact rational arithmetic, not rounded decimals.
• S157 · premise_and_result_audit — Recompute Exp term15 factorial step5 with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Exp term15 factorial step4 with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Exp term15 factorial step3 with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Exp term15 factorial step2 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Recompute Exp term15 factorial step1 with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Recompute Exp cumulative sum through14 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Exp term14 power over factorial with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S149 · premise_and_result_audit — Recompute Exp term14 factorial step14 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Exp term14 factorial step13 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Exp term14 factorial step12 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Exp term14 factorial step11 with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Exp term14 factorial step10 with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Exp term14 factorial step9 with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Exp term14 factorial step8 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Exp term14 factorial step7 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Exp term14 factorial step6 with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Exp term14 factorial step5 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Exp term14 factorial step4 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Exp term14 factorial step3 with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Exp term14 factorial step2 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Exp term14 factorial step1 with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Exp cumulative sum through13 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Exp term13 power over factorial with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute Exp term13 factorial step13 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Exp term13 factorial step12 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Exp term13 factorial step11 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Exp term13 factorial step10 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Exp term13 factorial step9 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Exp term13 factorial step8 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Exp term13 factorial step7 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Exp term13 factorial step6 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Exp term13 factorial step5 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Exp term13 factorial step4 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Exp term13 factorial step3 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Exp term13 factorial step2 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Exp term13 factorial step1 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Exp cumulative sum through12 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Exp term12 power over factorial with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Recompute Exp term12 factorial step12 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Exp term12 factorial step11 with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Exp term12 factorial step10 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Exp term12 factorial step9 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Exp term12 factorial step8 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Exp term12 factorial step7 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Exp term12 factorial step6 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Exp term12 factorial step5 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Exp term12 factorial step4 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Exp term12 factorial step3 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Exp term12 factorial step2 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Exp term12 factorial step1 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Exp cumulative sum through11 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Exp term11 power over factorial with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Recompute Exp term11 factorial step11 with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Exp term11 factorial step10 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Exp term11 factorial step9 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Exp term11 factorial step8 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Exp term11 factorial step7 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Exp term11 factorial step6 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Exp term11 factorial step5 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Exp term11 factorial step4 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Exp term11 factorial step3 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Exp term11 factorial step2 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Exp term11 factorial step1 with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Exp cumulative sum through10 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Exp term10 power over factorial with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Recompute Exp term10 factorial step10 with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Exp term10 factorial step9 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Exp term10 factorial step8 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Exp term10 factorial step7 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Exp term10 factorial step6 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Exp term10 factorial step5 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Exp term10 factorial step4 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Exp term10 factorial step3 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Exp term10 factorial step2 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Exp term10 factorial step1 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Exp cumulative sum through9 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Exp term9 power over factorial with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute Exp term9 factorial step9 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Exp term9 factorial step8 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Exp term9 factorial step7 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Exp term9 factorial step6 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Exp term9 factorial step5 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Exp term9 factorial step4 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Exp term9 factorial step3 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Exp term9 factorial step2 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Exp term9 factorial step1 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Exp cumulative sum through8 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Exp term8 power over factorial with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Recompute Exp term8 factorial step8 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Exp term8 factorial step7 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Exp term8 factorial step6 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Exp term8 factorial step5 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Exp term8 factorial step4 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Exp term8 factorial step3 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Exp term8 factorial step2 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Exp term8 factorial step1 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Exp cumulative sum through7 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Exp term7 power over factorial with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Recompute Exp term7 factorial step7 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Exp term7 factorial step6 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Exp term7 factorial step5 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Exp term7 factorial step4 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Exp term7 factorial step3 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Exp term7 factorial step2 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Exp term7 factorial step1 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Exp cumulative sum through6 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Exp term6 power over factorial with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute Exp term6 factorial step6 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Exp term6 factorial step5 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Exp term6 factorial step4 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Exp term6 factorial step3 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Exp term6 factorial step2 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Exp term6 factorial step1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Exp cumulative sum through5 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Exp term5 power over factorial with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Exp term5 factorial step5 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Exp term5 factorial step4 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Exp term5 factorial step3 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Exp term5 factorial step2 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Exp term5 factorial step1 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Exp cumulative sum through4 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Exp term4 power over factorial with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute Exp term4 factorial step4 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Exp term4 factorial step3 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Exp term4 factorial step2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Exp term4 factorial step1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Exp cumulative sum through3 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Exp term3 power over factorial with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Exp term3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Exp term3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Exp term3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Exp cumulative sum through2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Exp term2 power over factorial with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute Exp term2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Exp term2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Exp cumulative sum through1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Exp term1 power over factorial with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Exp term1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Exp cumulative sum through0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Exp term0 power over factorial with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.