第 24241 題：實指數函數

原題與已知條件

E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[16,17],"cutoff":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[16,17],"cutoff":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-real-exponential-function

逐步推導

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

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

source roles = [0]

derived = [1]

(16/17)^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

(16/17)^1

Exp ORIGINAL x power1 = 16/17

Exp term1 power over factorial := div(16/17, 1)

Exp term1 power over factorial = 16/17

Exp cumulative sum through1 := sum(1, 16/17)

Exp cumulative sum through1 = 33/17

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(16/17)^2

Exp ORIGINAL x power2 = 256/289

Exp term2 power over factorial := div(256/289, 2)

Exp term2 power over factorial = 128/289

Exp cumulative sum through2 := sum(33/17, 128/289)

Exp cumulative sum through2 = 689/289

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

(16/17)^3

Exp ORIGINAL x power3 = 4096/4913

Exp term3 power over factorial := div(4096/4913, 6)

Exp term3 power over factorial = 2048/14739

Exp cumulative sum through3 := sum(689/289, 2048/14739)

Exp cumulative sum through3 = 37187/14739

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

(16/17)^4

Exp ORIGINAL x power4 = 65536/83521

Exp term4 power over factorial := div(65536/83521, 24)

Exp term4 power over factorial = 8192/250563

Exp cumulative sum through4 := sum(37187/14739, 8192/250563)

Exp cumulative sum through4 = 213457/83521

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

(16/17)^5

Exp ORIGINAL x power5 = 1048576/1419857

Exp term5 power over factorial := div(1048576/1419857, 120)

Exp term5 power over factorial = 131072/21297855

Exp cumulative sum through5 := sum(213457/83521, 131072/21297855)

Exp cumulative sum through5 = 54562607/21297855

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

(16/17)^6

Exp ORIGINAL x power6 = 16777216/24137569

Exp term6 power over factorial := div(16777216/24137569, 720)

Exp term6 power over factorial = 1048576/1086190605

Exp cumulative sum through6 := sum(54562607/21297855, 1048576/1086190605)

Exp cumulative sum through6 = 2783741533/1086190605

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

(16/17)^7

Exp ORIGINAL x power7 = 268435456/410338673

Exp term7 power over factorial := div(268435456/410338673, 5040)

Exp term7 power over factorial = 16777216/129256681995

Exp cumulative sum through7 := sum(2783741533/1086190605, 16777216/129256681995)

Exp cumulative sum through7 = 110427339881/43085560665

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

(16/17)^8

Exp ORIGINAL x power8 = 4294967296/6975757441

Exp term8 power over factorial := div(4294967296/6975757441, 40320)

Exp term8 power over factorial = 33554432/2197363593915

Exp cumulative sum through8 := sum(110427339881/43085560665, 33554432/2197363593915)

Exp cumulative sum through8 = 5631827888363/2197363593915

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

(16/17)^9

Exp ORIGINAL x power9 = 68719476736/118587876497

Exp term9 power over factorial := div(68719476736/118587876497, 362880)

Exp term9 power over factorial = 536870912/336196629868995

Exp cumulative sum through9 := sum(5631827888363/2197363593915, 536870912/336196629868995)

Exp cumulative sum through9 = 861670203790451/336196629868995

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

(16/17)^10

Exp ORIGINAL x power10 = 1099511627776/2015993900449

Exp term10 power over factorial := div(1099511627776/2015993900449, 3628800)

Exp term10 power over factorial = 4294967296/28576713538864575

Exp cumulative sum through10 := sum(861670203790451/336196629868995, 4294967296/28576713538864575)

Exp cumulative sum through10 = 24413990539051877/9525571179621525

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

(16/17)^11

Exp ORIGINAL x power11 = 17592186044416/34271896307633

Exp term11 power over factorial := div(17592186044416/34271896307633, 39916800)

Exp term11 power over factorial = 68719476736/5343845431767675525

Exp cumulative sum through11 := sum(24413990539051877/9525571179621525, 68719476736/5343845431767675525)

Exp cumulative sum through11 = 13696248761127579733/5343845431767675525

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

(16/17)^12

Exp ORIGINAL x power12 = 281474976710656/582622237229761

Exp term12 power over factorial := div(281474976710656/582622237229761, 479001600)

Exp term12 power over factorial = 274877906944/272536117020151451775

Exp cumulative sum through12 := sum(13696248761127579733/5343845431767675525, 274877906944/272536117020151451775)

Exp cumulative sum through12 = 63500789735671315757/24776010638195586525

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

(16/17)^13

Exp ORIGINAL x power13 = 4503599627370496/9904578032905937

Exp term13 power over factorial := div(4503599627370496/9904578032905937, 6227020800)

Exp term13 power over factorial = 4398046511104/60230481861453470842275

Exp cumulative sum through13 := sum(63500789735671315757/24776010638195586525, 4398046511104/60230481861453470842275)

Exp cumulative sum through13 = 51456806617271671705457/20076827287151156947425

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

(16/17)^14

Exp ORIGINAL x power14 = 72057594037927936/168377826559400929

Exp term14 power over factorial := div(72057594037927936/168377826559400929, 87178291200)

Exp term14 power over factorial = 35184372088832/7167427341512963030230725

Exp cumulative sum through14 := sum(51456806617271671705457/20076827287151156947425, 35184372088832/7167427341512963030230725)

Exp cumulative sum through14 = 18370079962401171170936981/7167427341512963030230725

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

(16/17)^15

Exp ORIGINAL x power15 = 1152921504606846976/2862423051509815793

Exp term15 power over factorial := div(1152921504606846976/2862423051509815793, 1307674368000)

Exp term15 power over factorial = 562949953421312/1827693972085805572708834875

Exp cumulative sum through15 := sum(18370079962401171170936981/7167427341512963030230725, 562949953421312/1827693972085805572708834875)

Exp cumulative sum through15 = 360336183877912430657103959/140591844006600428669910375

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

(16/17)^16

Exp ORIGINAL x power16 = 18446744073709551616/48661191875666868481

Exp term16 power over factorial := div(18446744073709551616/48661191875666868481, 20922789888000)

Exp term16 power over factorial = 562949953421312/31070797525458694736050192875

Exp cumulative sum through16 := sum(360336183877912430657103959/140591844006600428669910375, 562949953421312/31070797525458694736050192875)

Exp cumulative sum through16 = 2413160504152097276520405947/941539318953293779880308875

Exp term17 factorial step1 := mul(1, 1)

Exp term17 factorial step1 = 1

Exp term17 factorial step2 := mul(1, 2)

Exp term17 factorial step2 = 2

Exp term17 factorial step3 := mul(2, 3)

Exp term17 factorial step3 = 6

Exp term17 factorial step4 := mul(6, 4)

Exp term17 factorial step4 = 24

Exp term17 factorial step5 := mul(24, 5)

Exp term17 factorial step5 = 120

Exp term17 factorial step6 := mul(120, 6)

Exp term17 factorial step6 = 720

Exp term17 factorial step7 := mul(720, 7)

Exp term17 factorial step7 = 5040

Exp term17 factorial step8 := mul(5040, 8)

Exp term17 factorial step8 = 40320

Exp term17 factorial step9 := mul(40320, 9)

Exp term17 factorial step9 = 362880

Exp term17 factorial step10 := mul(362880, 10)

Exp term17 factorial step10 = 3628800

Exp term17 factorial step11 := mul(3628800, 11)

Exp term17 factorial step11 = 39916800

Exp term17 factorial step12 := mul(39916800, 12)

Exp term17 factorial step12 = 479001600

Exp term17 factorial step13 := mul(479001600, 13)

Exp term17 factorial step13 = 6227020800

Exp term17 factorial step14 := mul(6227020800, 14)

Exp term17 factorial step14 = 87178291200

Exp term17 factorial step15 := mul(87178291200, 15)

Exp term17 factorial step15 = 1307674368000

Exp term17 factorial step16 := mul(1307674368000, 16)

Exp term17 factorial step16 = 20922789888000

Exp term17 factorial step17 := mul(20922789888000, 17)

Exp term17 factorial step17 = 355687428096000

(16/17)^17

Exp ORIGINAL x power17 = 295147905179352825856/827240261886336764177

Exp term17 power over factorial := div(295147905179352825856/827240261886336764177, 355687428096000)

Exp term17 power over factorial = 9007199254740992/8979460484857562778718505740875

Exp cumulative sum through17 := sum(2413160504152097276520405947/941539318953293779880308875, 9007199254740992/8979460484857562778718505740875)

Exp cumulative sum through17 = 23014311728098560733374366257531/8979460484857562778718505740875

Exp term18 factorial step1 := mul(1, 1)

Exp term18 factorial step1 = 1

Exp term18 factorial step2 := mul(1, 2)

Exp term18 factorial step2 = 2

Exp term18 factorial step3 := mul(2, 3)

Exp term18 factorial step3 = 6

Exp term18 factorial step4 := mul(6, 4)

Exp term18 factorial step4 = 24

Exp term18 factorial step5 := mul(24, 5)

Exp term18 factorial step5 = 120

Exp term18 factorial step6 := mul(120, 6)

Exp term18 factorial step6 = 720

Exp term18 factorial step7 := mul(720, 7)

Exp term18 factorial step7 = 5040

Exp term18 factorial step8 := mul(5040, 8)

Exp term18 factorial step8 = 40320

Exp term18 factorial step9 := mul(40320, 9)

Exp term18 factorial step9 = 362880

Exp term18 factorial step10 := mul(362880, 10)

Exp term18 factorial step10 = 3628800

Exp term18 factorial step11 := mul(3628800, 11)

Exp term18 factorial step11 = 39916800

Exp term18 factorial step12 := mul(39916800, 12)

Exp term18 factorial step12 = 479001600

Exp term18 factorial step13 := mul(479001600, 13)

Exp term18 factorial step13 = 6227020800

Exp term18 factorial step14 := mul(6227020800, 14)

Exp term18 factorial step14 = 87178291200

Exp term18 factorial step15 := mul(87178291200, 15)

Exp term18 factorial step15 = 1307674368000

Exp term18 factorial step16 := mul(1307674368000, 16)

Exp term18 factorial step16 = 20922789888000

Exp term18 factorial step17 := mul(20922789888000, 17)

Exp term18 factorial step17 = 355687428096000

Exp term18 factorial step18 := mul(355687428096000, 18)

Exp term18 factorial step18 = 6402373705728000

(16/17)^18

Exp ORIGINAL x power18 = 4722366482869645213696/14063084452067724991009

Exp term18 power over factorial := div(4722366482869645213696/14063084452067724991009, 6402373705728000)

Exp term18 power over factorial = 72057594037927936/1373857454183207105143931378353875

Exp cumulative sum through18 := sum(23014311728098560733374366257531/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875)

Exp cumulative sum through18 = 3521189694399079864263872075330179/1373857454183207105143931378353875

Exp term19 factorial step1 := mul(1, 1)

Exp term19 factorial step1 = 1

Exp term19 factorial step2 := mul(1, 2)

Exp term19 factorial step2 = 2

Exp term19 factorial step3 := mul(2, 3)

Exp term19 factorial step3 = 6

Exp term19 factorial step4 := mul(6, 4)

Exp term19 factorial step4 = 24

Exp term19 factorial step5 := mul(24, 5)

Exp term19 factorial step5 = 120

Exp term19 factorial step6 := mul(120, 6)

Exp term19 factorial step6 = 720

Exp term19 factorial step7 := mul(720, 7)

Exp term19 factorial step7 = 5040

Exp term19 factorial step8 := mul(5040, 8)

Exp term19 factorial step8 = 40320

Exp term19 factorial step9 := mul(40320, 9)

Exp term19 factorial step9 = 362880

Exp term19 factorial step10 := mul(362880, 10)

Exp term19 factorial step10 = 3628800

Exp term19 factorial step11 := mul(3628800, 11)

Exp term19 factorial step11 = 39916800

Exp term19 factorial step12 := mul(39916800, 12)

Exp term19 factorial step12 = 479001600

Exp term19 factorial step13 := mul(479001600, 13)

Exp term19 factorial step13 = 6227020800

Exp term19 factorial step14 := mul(6227020800, 14)

Exp term19 factorial step14 = 87178291200

Exp term19 factorial step15 := mul(87178291200, 15)

Exp term19 factorial step15 = 1307674368000

Exp term19 factorial step16 := mul(1307674368000, 16)

Exp term19 factorial step16 = 20922789888000

Exp term19 factorial step17 := mul(20922789888000, 17)

Exp term19 factorial step17 = 355687428096000

Exp term19 factorial step18 := mul(355687428096000, 18)

Exp term19 factorial step18 = 6402373705728000

Exp term19 factorial step19 := mul(6402373705728000, 19)

Exp term19 factorial step19 = 121645100408832000

(16/17)^19

Exp ORIGINAL x power19 = 75557863725914323419136/239072435685151324847153

Exp term19 power over factorial := div(75557863725914323419136/239072435685151324847153, 121645100408832000)

Exp term19 power over factorial = 1152921504606846976/443755957701175894961489835208301625

Exp cumulative sum through19 := sum(3521189694399079864263872075330179/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625)

Exp cumulative sum through19 = 379114757096967599103384061646164931/147918652567058631653829945069433875

Exp term20 factorial step1 := mul(1, 1)

Exp term20 factorial step1 = 1

Exp term20 factorial step2 := mul(1, 2)

Exp term20 factorial step2 = 2

Exp term20 factorial step3 := mul(2, 3)

Exp term20 factorial step3 = 6

Exp term20 factorial step4 := mul(6, 4)

Exp term20 factorial step4 = 24

Exp term20 factorial step5 := mul(24, 5)

Exp term20 factorial step5 = 120

Exp term20 factorial step6 := mul(120, 6)

Exp term20 factorial step6 = 720

Exp term20 factorial step7 := mul(720, 7)

Exp term20 factorial step7 = 5040

Exp term20 factorial step8 := mul(5040, 8)

Exp term20 factorial step8 = 40320

Exp term20 factorial step9 := mul(40320, 9)

Exp term20 factorial step9 = 362880

Exp term20 factorial step10 := mul(362880, 10)

Exp term20 factorial step10 = 3628800

Exp term20 factorial step11 := mul(3628800, 11)

Exp term20 factorial step11 = 39916800

Exp term20 factorial step12 := mul(39916800, 12)

Exp term20 factorial step12 = 479001600

Exp term20 factorial step13 := mul(479001600, 13)

Exp term20 factorial step13 = 6227020800

Exp term20 factorial step14 := mul(6227020800, 14)

Exp term20 factorial step14 = 87178291200

Exp term20 factorial step15 := mul(87178291200, 15)

Exp term20 factorial step15 = 1307674368000

Exp term20 factorial step16 := mul(1307674368000, 16)

Exp term20 factorial step16 = 20922789888000

Exp term20 factorial step17 := mul(20922789888000, 17)

Exp term20 factorial step17 = 355687428096000

Exp term20 factorial step18 := mul(355687428096000, 18)

Exp term20 factorial step18 = 6402373705728000

Exp term20 factorial step19 := mul(6402373705728000, 19)

Exp term20 factorial step19 = 121645100408832000

Exp term20 factorial step20 := mul(121645100408832000, 20)

Exp term20 factorial step20 = 2432902008176640000

(16/17)^20

Exp ORIGINAL x power20 = 1208925819614629174706176/4064231406647572522401601

Exp term20 power over factorial := div(1208925819614629174706176/4064231406647572522401601, 2432902008176640000)

Exp term20 power over factorial = 4611686018427387904/37719256404599951071726635992705638125

Exp cumulative sum through20 := sum(379114757096967599103384061646164931/147918652567058631653829945069433875, 4611686018427387904/37719256404599951071726635992705638125)

Exp cumulative sum through20 = 8788569369066067070543147430745404119/3429023309509086461066057817518694375

source roles = [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]]

derived = [8788569369066067070543147430745404119/3429023309509086461066057817518694375, 否, finite sum is not a general infinite-value certificate]

逐步反向核對

• S276 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S275 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through20，不以小數近似替代。
• S274 · premise_and_result_audit — 以精確有理數重算 Exp term20 power over factorial，不以小數近似替代。
• S273 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S272 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step20，不以小數近似替代。
• S271 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step19，不以小數近似替代。
• S270 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step18，不以小數近似替代。
• S269 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step17，不以小數近似替代。
• S268 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step16，不以小數近似替代。
• S267 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step15，不以小數近似替代。
• S266 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step14，不以小數近似替代。
• S265 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step13，不以小數近似替代。
• S264 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step12，不以小數近似替代。
• S263 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step11，不以小數近似替代。
• S262 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step10，不以小數近似替代。
• S261 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step9，不以小數近似替代。
• S260 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step8，不以小數近似替代。
• S259 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step7，不以小數近似替代。
• S258 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step6，不以小數近似替代。
• S257 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step5，不以小數近似替代。
• S256 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step4，不以小數近似替代。
• S255 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step3，不以小數近似替代。
• S254 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step2，不以小數近似替代。
• S253 · premise_and_result_audit — 以精確有理數重算 Exp term20 factorial step1，不以小數近似替代。
• S252 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through19，不以小數近似替代。
• S251 · premise_and_result_audit — 以精確有理數重算 Exp term19 power over factorial，不以小數近似替代。
• S250 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S249 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step19，不以小數近似替代。
• S248 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step18，不以小數近似替代。
• S247 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step17，不以小數近似替代。
• S246 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step16，不以小數近似替代。
• S245 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step15，不以小數近似替代。
• S244 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step14，不以小數近似替代。
• S243 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step13，不以小數近似替代。
• S242 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step12，不以小數近似替代。
• S241 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step11，不以小數近似替代。
• S240 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step10，不以小數近似替代。
• S239 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step9，不以小數近似替代。
• S238 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step8，不以小數近似替代。
• S237 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step7，不以小數近似替代。
• S236 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step6，不以小數近似替代。
• S235 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step5，不以小數近似替代。
• S234 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step4，不以小數近似替代。
• S233 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step3，不以小數近似替代。
• S232 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step2，不以小數近似替代。
• S231 · premise_and_result_audit — 以精確有理數重算 Exp term19 factorial step1，不以小數近似替代。
• S230 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through18，不以小數近似替代。
• S229 · premise_and_result_audit — 以精確有理數重算 Exp term18 power over factorial，不以小數近似替代。
• S228 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S227 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step18，不以小數近似替代。
• S226 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step17，不以小數近似替代。
• S225 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step16，不以小數近似替代。
• S224 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step15，不以小數近似替代。
• S223 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step14，不以小數近似替代。
• S222 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step13，不以小數近似替代。
• S221 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step12，不以小數近似替代。
• S220 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step11，不以小數近似替代。
• S219 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step10，不以小數近似替代。
• S218 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step9，不以小數近似替代。
• S217 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step8，不以小數近似替代。
• S216 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step7，不以小數近似替代。
• S215 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step6，不以小數近似替代。
• S214 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step5，不以小數近似替代。
• S213 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step4，不以小數近似替代。
• S212 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step3，不以小數近似替代。
• S211 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step2，不以小數近似替代。
• S210 · premise_and_result_audit — 以精確有理數重算 Exp term18 factorial step1，不以小數近似替代。
• S209 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through17，不以小數近似替代。
• S208 · premise_and_result_audit — 以精確有理數重算 Exp term17 power over factorial，不以小數近似替代。
• S207 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S206 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step17，不以小數近似替代。
• S205 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step16，不以小數近似替代。
• S204 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step15，不以小數近似替代。
• S203 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step14，不以小數近似替代。
• S202 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step13，不以小數近似替代。
• S201 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step12，不以小數近似替代。
• S200 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step11，不以小數近似替代。
• S199 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step10，不以小數近似替代。
• S198 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step9，不以小數近似替代。
• S197 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step8，不以小數近似替代。
• S196 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step7，不以小數近似替代。
• S195 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step6，不以小數近似替代。
• S194 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step5，不以小數近似替代。
• S193 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step4，不以小數近似替代。
• S192 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step3，不以小數近似替代。
• S191 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step2，不以小數近似替代。
• S190 · premise_and_result_audit — 以精確有理數重算 Exp term17 factorial step1，不以小數近似替代。
• S189 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through16，不以小數近似替代。
• S188 · premise_and_result_audit — 以精確有理數重算 Exp term16 power over factorial，不以小數近似替代。
• S187 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S186 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step16，不以小數近似替代。
• S185 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step15，不以小數近似替代。
• S184 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step14，不以小數近似替代。
• S183 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step13，不以小數近似替代。
• S182 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step12，不以小數近似替代。
• S181 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step11，不以小數近似替代。
• S180 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step10，不以小數近似替代。
• S179 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step9，不以小數近似替代。
• S178 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step8，不以小數近似替代。
• S177 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step7，不以小數近似替代。
• S176 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step6，不以小數近似替代。
• S175 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step5，不以小數近似替代。
• S174 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step4，不以小數近似替代。
• S173 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step3，不以小數近似替代。
• S172 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step2，不以小數近似替代。
• S171 · premise_and_result_audit — 以精確有理數重算 Exp term16 factorial step1，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through15，不以小數近似替代。
• S169 · premise_and_result_audit — 以精確有理數重算 Exp term15 power over factorial，不以小數近似替代。
• S168 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S167 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step15，不以小數近似替代。
• S166 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step14，不以小數近似替代。
• S165 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step13，不以小數近似替代。
• S164 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step12，不以小數近似替代。
• S163 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step11，不以小數近似替代。
• S162 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step10，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step9，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step8，不以小數近似替代。
• S159 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step7，不以小數近似替代。
• S158 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step6，不以小數近似替代。
• S157 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step5，不以小數近似替代。
• S156 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step4，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step3，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step2，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 Exp term15 factorial step1，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through14，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Exp term14 power over factorial，不以小數近似替代。
• S150 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S149 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step14，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step13，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step12，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step11，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step10，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step9，不以小數近似替代。
• S143 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step8，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step7，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step6，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step5，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step4，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step3，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step2，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 Exp term14 factorial step1，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through13，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Exp term13 power over factorial，不以小數近似替代。
• S133 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S132 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step13，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step12，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step11，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step10，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step9，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step8，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step7，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step6，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step5，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step4，不以小數近似替代。
• S122 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step3，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step2，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step1，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through12，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Exp term12 power over factorial，不以小數近似替代。
• S117 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S116 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step12，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step11，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step10，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step9，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step8，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step7，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step6，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step5，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step4，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step3，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step2，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through11，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Exp term11 power over factorial，不以小數近似替代。
• S102 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S101 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step11，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step10，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step9，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step8，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step7，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step6，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step5，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step4，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step3，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step2，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step1，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through10，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Exp term10 power over factorial，不以小數近似替代。
• S088 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S087 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step10，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step9，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step8，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step7，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step6，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step5，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step4，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step3，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step2，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step1，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through9，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Exp term9 power over factorial，不以小數近似替代。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step9，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step8，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step7，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step6，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step5，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step4，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step3，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step2，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step1，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through8，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Exp term8 power over factorial，不以小數近似替代。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step8，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step7，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step6，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step5，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step4，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step3，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step2，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step1，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through7，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Exp term7 power over factorial，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step7，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step6，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step5，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step4，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step3，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step2，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step1，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through6，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Exp term6 power over factorial，不以小數近似替代。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step6，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step5，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step4，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step3，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step2，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through5，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Exp term5 power over factorial，不以小數近似替代。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step5，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step4，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through4，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Exp term4 power over factorial，不以小數近似替代。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step4，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step3，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through3，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Exp term3 power over factorial，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Exp term2 power over factorial，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Exp term1 power over factorial，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 Exp term1 factorial step1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Exp term0 power over factorial，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(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, 否, finite_partial_sum_is_not_the_infinite_exponential_value)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。