Question 24242: 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=17/18, cutoff=21, 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=17/18, cutoff=21, 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]

(17/18)^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

(17/18)^1

Exp ORIGINAL x power1 = 17/18

Exp term1 power over factorial := div(17/18, 1)

Exp term1 power over factorial = 17/18

Exp cumulative sum through1 := sum(1, 17/18)

Exp cumulative sum through1 = 35/18

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(17/18)^2

Exp ORIGINAL x power2 = 289/324

Exp term2 power over factorial := div(289/324, 2)

Exp term2 power over factorial = 289/648

Exp cumulative sum through2 := sum(35/18, 289/648)

Exp cumulative sum through2 = 1549/648

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

(17/18)^3

Exp ORIGINAL x power3 = 4913/5832

Exp term3 power over factorial := div(4913/5832, 6)

Exp term3 power over factorial = 4913/34992

Exp cumulative sum through3 := sum(1549/648, 4913/34992)

Exp cumulative sum through3 = 88559/34992

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

(17/18)^4

Exp ORIGINAL x power4 = 83521/104976

Exp term4 power over factorial := div(83521/104976, 24)

Exp term4 power over factorial = 83521/2519424

Exp cumulative sum through4 := sum(88559/34992, 83521/2519424)

Exp cumulative sum through4 = 6459769/2519424

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

(17/18)^5

Exp ORIGINAL x power5 = 1419857/1889568

Exp term5 power over factorial := div(1419857/1889568, 120)

Exp term5 power over factorial = 1419857/226748160

Exp cumulative sum through5 := sum(6459769/2519424, 1419857/226748160)

Exp cumulative sum through5 = 582799067/226748160

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

(17/18)^6

Exp ORIGINAL x power6 = 24137569/34012224

Exp term6 power over factorial := div(24137569/34012224, 720)

Exp term6 power over factorial = 24137569/24488801280

Exp cumulative sum through6 := sum(582799067/226748160, 24137569/24488801280)

Exp cumulative sum through6 = 12593287361/4897760256

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

(17/18)^7

Exp ORIGINAL x power7 = 410338673/612220032

Exp term7 power over factorial := div(410338673/612220032, 5040)

Exp term7 power over factorial = 410338673/3085588961280

Exp cumulative sum through7 := sum(12593287361/4897760256, 410338673/3085588961280)

Exp cumulative sum through7 = 7934181376103/3085588961280

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

(17/18)^8

Exp ORIGINAL x power8 = 6975757441/11019960576

Exp term8 power over factorial := div(6975757441/11019960576, 40320)

Exp term8 power over factorial = 6975757441/444324810424320

Exp cumulative sum through8 := sum(7934181376103/3085588961280, 6975757441/444324810424320)

Exp cumulative sum through8 = 163218441988039/63474972917760

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

(17/18)^9

Exp ORIGINAL x power9 = 118587876497/198359290368

Exp term9 power over factorial := div(118587876497/198359290368, 362880)

Exp term9 power over factorial = 118587876497/71980619288739840

Exp cumulative sum through9 := sum(163218441988039/63474972917760, 118587876497/71980619288739840)

Exp cumulative sum through9 = 185089831802312723/71980619288739840

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

(17/18)^10

Exp ORIGINAL x power10 = 2015993900449/3570467226624

Exp term10 power over factorial := div(2015993900449/3570467226624, 3628800)

Exp term10 power over factorial = 2015993900449/12956511471973171200

Exp cumulative sum through10 := sum(185089831802312723/71980619288739840, 2015993900449/12956511471973171200)

Exp cumulative sum through10 = 33316171740410190589/12956511471973171200

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

(17/18)^11

Exp ORIGINAL x power11 = 34271896307633/64268410079232

Exp term11 power over factorial := div(34271896307633/64268410079232, 39916800)

Exp term11 power over factorial = 34271896307633/2565389271450687897600

Exp cumulative sum through11 := sum(33316171740410190589/12956511471973171200, 34271896307633/2565389271450687897600)

Exp cumulative sum through11 = 1319320407774622808851/513077854290137579520

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

(17/18)^12

Exp ORIGINAL x power12 = 582622237229761/1156831381426176

Exp term12 power over factorial := div(582622237229761/1156831381426176, 479001600)

Exp term12 power over factorial = 582622237229761/554124082633348585881600

Exp cumulative sum through12 := sum(1319320407774622808851/513077854290137579520, 582622237229761/554124082633348585881600)

Exp cumulative sum through12 = 1424866040979214870788841/554124082633348585881600

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

(17/18)^13

Exp ORIGINAL x power13 = 9904578032905937/20822964865671168

Exp term13 power over factorial := div(9904578032905937/20822964865671168, 6227020800)

Exp term13 power over factorial = 9904578032905937/129665035336203569096294400

Exp cumulative sum through13 := sum(1424866040979214870788841/554124082633348585881600, 9904578032905937/129665035336203569096294400)

Exp cumulative sum through13 = 333418653599040857797494731/129665035336203569096294400

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

(17/18)^14

Exp ORIGINAL x power14 = 168377826559400929/374813367582081024

Exp term14 power over factorial := div(168377826559400929/374813367582081024, 87178291200)

Exp term14 power over factorial = 168377826559400929/32675588904723299412266188800

Exp cumulative sum through14 := sum(333418653599040857797494731/129665035336203569096294400, 168377826559400929/32675588904723299412266188800)

Exp cumulative sum through14 = 84021500707126673991528073141/32675588904723299412266188800

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

(17/18)^15

Exp ORIGINAL x power15 = 2862423051509815793/6746640616477458432

Exp term15 power over factorial := div(2862423051509815793/6746640616477458432, 1307674368000)

Exp term15 power over factorial = 2862423051509815793/8822409004275290841311870976000

Exp cumulative sum through15 := sum(84021500707126673991528073141/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000)

Exp cumulative sum through15 = 462975616141368661240083460487/180049163352556955945140224000

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

(17/18)^16

Exp ORIGINAL x power16 = 48661191875666868481/121439531096594251776

Exp term16 power over factorial := div(48661191875666868481/121439531096594251776, 20922789888000)

Exp term16 power over factorial = 48661191875666868481/2540853793231283762297818841088000

Exp cumulative sum through16 := sum(462975616141368661240083460487/180049163352556955945140224000, 48661191875666868481/2540853793231283762297818841088000)

Exp cumulative sum through16 = 261340475799481728344477338450441/101634151729251350491912753643520

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

(17/18)^17

Exp ORIGINAL x power17 = 827240261886336764177/2185911559738696531968

Exp term17 power over factorial := div(827240261886336764177/2185911559738696531968, 355687428096000)

Exp term17 power over factorial = 48661191875666868481/45735368278163107721360739139584000

Exp cumulative sum through17 := sum(261340475799481728344477338450441/101634151729251350491912753643520, 48661191875666868481/45735368278163107721360739139584000)

Exp cumulative sum through17 = 117603214109766826416206677969566931/45735368278163107721360739139584000

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

(17/18)^18

Exp ORIGINAL x power18 = 14063084452067724991009/39346408075296537575424

Exp term18 power over factorial := div(14063084452067724991009/39346408075296537575424, 6402373705728000)

Exp term18 power over factorial = 827240261886336764177/14818259322124846901720879481225216000

Exp cumulative sum through18 := sum(117603214109766826416206677969566931/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000)

Exp cumulative sum through18 = 38103441371564452586091225548476449821/14818259322124846901720879481225216000

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

(17/18)^19

Exp ORIGINAL x power19 = 239072435685151324847153/708235345355337676357632

Exp term19 power over factorial := div(239072435685151324847153/708235345355337676357632, 121645100408832000)

Exp term19 power over factorial = 14063084452067724991009/5067844688166697640388540782579023872000

Exp cumulative sum through19 := sum(38103441371564452586091225548476449821/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000)

Exp cumulative sum through19 = 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000

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

(17/18)^20

Exp ORIGINAL x power20 = 4064231406647572522401601/12748236216396078174437376

Exp term20 power over factorial := div(4064231406647572522401601/12748236216396078174437376, 2432902008176640000)

Exp term20 power over factorial = 239072435685151324847153/1824424087740011150539874681728448593920000

Exp cumulative sum through20 := sum(13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000)

Exp cumulative sum through20 = 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000

Exp term21 factorial step1 := mul(1, 1)

Exp term21 factorial step1 = 1

Exp term21 factorial step2 := mul(1, 2)

Exp term21 factorial step2 = 2

Exp term21 factorial step3 := mul(2, 3)

Exp term21 factorial step3 = 6

Exp term21 factorial step4 := mul(6, 4)

Exp term21 factorial step4 = 24

Exp term21 factorial step5 := mul(24, 5)

Exp term21 factorial step5 = 120

Exp term21 factorial step6 := mul(120, 6)

Exp term21 factorial step6 = 720

Exp term21 factorial step7 := mul(720, 7)

Exp term21 factorial step7 = 5040

Exp term21 factorial step8 := mul(5040, 8)

Exp term21 factorial step8 = 40320

Exp term21 factorial step9 := mul(40320, 9)

Exp term21 factorial step9 = 362880

Exp term21 factorial step10 := mul(362880, 10)

Exp term21 factorial step10 = 3628800

Exp term21 factorial step11 := mul(3628800, 11)

Exp term21 factorial step11 = 39916800

Exp term21 factorial step12 := mul(39916800, 12)

Exp term21 factorial step12 = 479001600

Exp term21 factorial step13 := mul(479001600, 13)

Exp term21 factorial step13 = 6227020800

Exp term21 factorial step14 := mul(6227020800, 14)

Exp term21 factorial step14 = 87178291200

Exp term21 factorial step15 := mul(87178291200, 15)

Exp term21 factorial step15 = 1307674368000

Exp term21 factorial step16 := mul(1307674368000, 16)

Exp term21 factorial step16 = 20922789888000

Exp term21 factorial step17 := mul(20922789888000, 17)

Exp term21 factorial step17 = 355687428096000

Exp term21 factorial step18 := mul(355687428096000, 18)

Exp term21 factorial step18 = 6402373705728000

Exp term21 factorial step19 := mul(6402373705728000, 19)

Exp term21 factorial step19 = 121645100408832000

Exp term21 factorial step20 := mul(121645100408832000, 20)

Exp term21 factorial step20 = 2432902008176640000

Exp term21 factorial step21 := mul(2432902008176640000, 21)

Exp term21 factorial step21 = 51090942171709440000

(17/18)^21

Exp ORIGINAL x power21 = 69091933913008732880827217/229468251895129407139872768

Exp term21 power over factorial := div(69091933913008732880827217/229468251895129407139872768, 51090942171709440000)

Exp term21 power over factorial = 4064231406647572522401601/689632305165724214904072629693353568501760000

Exp cumulative sum through21 := sum(426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 4064231406647572522401601/689632305165724214904072629693353568501760000)

Exp cumulative sum through21 = 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000

source roles = [17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000]]

derived = [354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, No, finite sum is not a general infinite-value certificate]

Reverse checks

• S300 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S299 · premise_and_result_audit — Recompute Exp cumulative sum through21 with exact rational arithmetic, not rounded decimals.
• S298 · premise_and_result_audit — Recompute Exp term21 power over factorial with exact rational arithmetic, not rounded decimals.
• S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S296 · premise_and_result_audit — Recompute Exp term21 factorial step21 with exact rational arithmetic, not rounded decimals.
• S295 · premise_and_result_audit — Recompute Exp term21 factorial step20 with exact rational arithmetic, not rounded decimals.
• S294 · premise_and_result_audit — Recompute Exp term21 factorial step19 with exact rational arithmetic, not rounded decimals.
• S293 · premise_and_result_audit — Recompute Exp term21 factorial step18 with exact rational arithmetic, not rounded decimals.
• S292 · premise_and_result_audit — Recompute Exp term21 factorial step17 with exact rational arithmetic, not rounded decimals.
• S291 · premise_and_result_audit — Recompute Exp term21 factorial step16 with exact rational arithmetic, not rounded decimals.
• S290 · premise_and_result_audit — Recompute Exp term21 factorial step15 with exact rational arithmetic, not rounded decimals.
• S289 · premise_and_result_audit — Recompute Exp term21 factorial step14 with exact rational arithmetic, not rounded decimals.
• S288 · premise_and_result_audit — Recompute Exp term21 factorial step13 with exact rational arithmetic, not rounded decimals.
• S287 · premise_and_result_audit — Recompute Exp term21 factorial step12 with exact rational arithmetic, not rounded decimals.
• S286 · premise_and_result_audit — Recompute Exp term21 factorial step11 with exact rational arithmetic, not rounded decimals.
• S285 · premise_and_result_audit — Recompute Exp term21 factorial step10 with exact rational arithmetic, not rounded decimals.
• S284 · premise_and_result_audit — Recompute Exp term21 factorial step9 with exact rational arithmetic, not rounded decimals.
• S283 · premise_and_result_audit — Recompute Exp term21 factorial step8 with exact rational arithmetic, not rounded decimals.
• S282 · premise_and_result_audit — Recompute Exp term21 factorial step7 with exact rational arithmetic, not rounded decimals.
• S281 · premise_and_result_audit — Recompute Exp term21 factorial step6 with exact rational arithmetic, not rounded decimals.
• S280 · premise_and_result_audit — Recompute Exp term21 factorial step5 with exact rational arithmetic, not rounded decimals.
• S279 · premise_and_result_audit — Recompute Exp term21 factorial step4 with exact rational arithmetic, not rounded decimals.
• S278 · premise_and_result_audit — Recompute Exp term21 factorial step3 with exact rational arithmetic, not rounded decimals.
• S277 · premise_and_result_audit — Recompute Exp term21 factorial step2 with exact rational arithmetic, not rounded decimals.
• S276 · premise_and_result_audit — Recompute Exp term21 factorial step1 with exact rational arithmetic, not rounded decimals.
• S275 · premise_and_result_audit — Recompute Exp cumulative sum through20 with exact rational arithmetic, not rounded decimals.
• S274 · premise_and_result_audit — Recompute Exp term20 power over factorial with exact rational arithmetic, not rounded decimals.
• S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S272 · premise_and_result_audit — Recompute Exp term20 factorial step20 with exact rational arithmetic, not rounded decimals.
• S271 · premise_and_result_audit — Recompute Exp term20 factorial step19 with exact rational arithmetic, not rounded decimals.
• S270 · premise_and_result_audit — Recompute Exp term20 factorial step18 with exact rational arithmetic, not rounded decimals.
• S269 · premise_and_result_audit — Recompute Exp term20 factorial step17 with exact rational arithmetic, not rounded decimals.
• S268 · premise_and_result_audit — Recompute Exp term20 factorial step16 with exact rational arithmetic, not rounded decimals.
• S267 · premise_and_result_audit — Recompute Exp term20 factorial step15 with exact rational arithmetic, not rounded decimals.
• S266 · premise_and_result_audit — Recompute Exp term20 factorial step14 with exact rational arithmetic, not rounded decimals.
• S265 · premise_and_result_audit — Recompute Exp term20 factorial step13 with exact rational arithmetic, not rounded decimals.
• S264 · premise_and_result_audit — Recompute Exp term20 factorial step12 with exact rational arithmetic, not rounded decimals.
• S263 · premise_and_result_audit — Recompute Exp term20 factorial step11 with exact rational arithmetic, not rounded decimals.
• S262 · premise_and_result_audit — Recompute Exp term20 factorial step10 with exact rational arithmetic, not rounded decimals.
• S261 · premise_and_result_audit — Recompute Exp term20 factorial step9 with exact rational arithmetic, not rounded decimals.
• S260 · premise_and_result_audit — Recompute Exp term20 factorial step8 with exact rational arithmetic, not rounded decimals.
• S259 · premise_and_result_audit — Recompute Exp term20 factorial step7 with exact rational arithmetic, not rounded decimals.
• S258 · premise_and_result_audit — Recompute Exp term20 factorial step6 with exact rational arithmetic, not rounded decimals.
• S257 · premise_and_result_audit — Recompute Exp term20 factorial step5 with exact rational arithmetic, not rounded decimals.
• S256 · premise_and_result_audit — Recompute Exp term20 factorial step4 with exact rational arithmetic, not rounded decimals.
• S255 · premise_and_result_audit — Recompute Exp term20 factorial step3 with exact rational arithmetic, not rounded decimals.
• S254 · premise_and_result_audit — Recompute Exp term20 factorial step2 with exact rational arithmetic, not rounded decimals.
• S253 · premise_and_result_audit — Recompute Exp term20 factorial step1 with exact rational arithmetic, not rounded decimals.
• S252 · premise_and_result_audit — Recompute Exp cumulative sum through19 with exact rational arithmetic, not rounded decimals.
• S251 · premise_and_result_audit — Recompute Exp term19 power over factorial with exact rational arithmetic, not rounded decimals.
• S250 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S249 · premise_and_result_audit — Recompute Exp term19 factorial step19 with exact rational arithmetic, not rounded decimals.
• S248 · premise_and_result_audit — Recompute Exp term19 factorial step18 with exact rational arithmetic, not rounded decimals.
• S247 · premise_and_result_audit — Recompute Exp term19 factorial step17 with exact rational arithmetic, not rounded decimals.
• S246 · premise_and_result_audit — Recompute Exp term19 factorial step16 with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Recompute Exp term19 factorial step15 with exact rational arithmetic, not rounded decimals.
• S244 · premise_and_result_audit — Recompute Exp term19 factorial step14 with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Recompute Exp term19 factorial step13 with exact rational arithmetic, not rounded decimals.
• S242 · premise_and_result_audit — Recompute Exp term19 factorial step12 with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Recompute Exp term19 factorial step11 with exact rational arithmetic, not rounded decimals.
• S240 · premise_and_result_audit — Recompute Exp term19 factorial step10 with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Recompute Exp term19 factorial step9 with exact rational arithmetic, not rounded decimals.
• S238 · premise_and_result_audit — Recompute Exp term19 factorial step8 with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Recompute Exp term19 factorial step7 with exact rational arithmetic, not rounded decimals.
• S236 · premise_and_result_audit — Recompute Exp term19 factorial step6 with exact rational arithmetic, not rounded decimals.
• S235 · premise_and_result_audit — Recompute Exp term19 factorial step5 with exact rational arithmetic, not rounded decimals.
• S234 · premise_and_result_audit — Recompute Exp term19 factorial step4 with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Recompute Exp term19 factorial step3 with exact rational arithmetic, not rounded decimals.
• S232 · premise_and_result_audit — Recompute Exp term19 factorial step2 with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Recompute Exp term19 factorial step1 with exact rational arithmetic, not rounded decimals.
• S230 · premise_and_result_audit — Recompute Exp cumulative sum through18 with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Recompute Exp term18 power over factorial with exact rational arithmetic, not rounded decimals.
• S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S227 · premise_and_result_audit — Recompute Exp term18 factorial step18 with exact rational arithmetic, not rounded decimals.
• S226 · premise_and_result_audit — Recompute Exp term18 factorial step17 with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Recompute Exp term18 factorial step16 with exact rational arithmetic, not rounded decimals.
• S224 · premise_and_result_audit — Recompute Exp term18 factorial step15 with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Recompute Exp term18 factorial step14 with exact rational arithmetic, not rounded decimals.
• S222 · premise_and_result_audit — Recompute Exp term18 factorial step13 with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Recompute Exp term18 factorial step12 with exact rational arithmetic, not rounded decimals.
• S220 · premise_and_result_audit — Recompute Exp term18 factorial step11 with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Recompute Exp term18 factorial step10 with exact rational arithmetic, not rounded decimals.
• S218 · premise_and_result_audit — Recompute Exp term18 factorial step9 with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Recompute Exp term18 factorial step8 with exact rational arithmetic, not rounded decimals.
• S216 · premise_and_result_audit — Recompute Exp term18 factorial step7 with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Recompute Exp term18 factorial step6 with exact rational arithmetic, not rounded decimals.
• S214 · premise_and_result_audit — Recompute Exp term18 factorial step5 with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Recompute Exp term18 factorial step4 with exact rational arithmetic, not rounded decimals.
• S212 · premise_and_result_audit — Recompute Exp term18 factorial step3 with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Recompute Exp term18 factorial step2 with exact rational arithmetic, not rounded decimals.
• S210 · premise_and_result_audit — Recompute Exp term18 factorial step1 with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Recompute Exp cumulative sum through17 with exact rational arithmetic, not rounded decimals.
• S208 · premise_and_result_audit — Recompute Exp term17 power over factorial with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S206 · premise_and_result_audit — Recompute Exp term17 factorial step17 with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Recompute Exp term17 factorial step16 with exact rational arithmetic, not rounded decimals.
• S204 · premise_and_result_audit — Recompute Exp term17 factorial step15 with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Recompute Exp term17 factorial step14 with exact rational arithmetic, not rounded decimals.
• S202 · premise_and_result_audit — Recompute Exp term17 factorial step13 with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Recompute Exp term17 factorial step12 with exact rational arithmetic, not rounded decimals.
• S200 · premise_and_result_audit — Recompute Exp term17 factorial step11 with exact rational arithmetic, not rounded decimals.
• S199 · premise_and_result_audit — Recompute Exp term17 factorial step10 with exact rational arithmetic, not rounded decimals.
• S198 · premise_and_result_audit — Recompute Exp term17 factorial step9 with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Recompute Exp term17 factorial step8 with exact rational arithmetic, not rounded decimals.
• S196 · premise_and_result_audit — Recompute Exp term17 factorial step7 with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Recompute Exp term17 factorial step6 with exact rational arithmetic, not rounded decimals.
• S194 · premise_and_result_audit — Recompute Exp term17 factorial step5 with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Recompute Exp term17 factorial step4 with exact rational arithmetic, not rounded decimals.
• S192 · premise_and_result_audit — Recompute Exp term17 factorial step3 with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Recompute Exp term17 factorial step2 with exact rational arithmetic, not rounded decimals.
• S190 · premise_and_result_audit — Recompute Exp term17 factorial step1 with exact rational arithmetic, not rounded decimals.
• 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

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

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.