Question 24245: 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=20/21, cutoff=24, 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=20/21, cutoff=24, 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]

(20/21)^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

(20/21)^1

Exp ORIGINAL x power1 = 20/21

Exp term1 power over factorial := div(20/21, 1)

Exp term1 power over factorial = 20/21

Exp cumulative sum through1 := sum(1, 20/21)

Exp cumulative sum through1 = 41/21

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(20/21)^2

Exp ORIGINAL x power2 = 400/441

Exp term2 power over factorial := div(400/441, 2)

Exp term2 power over factorial = 200/441

Exp cumulative sum through2 := sum(41/21, 200/441)

Exp cumulative sum through2 = 1061/441

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

(20/21)^3

Exp ORIGINAL x power3 = 8000/9261

Exp term3 power over factorial := div(8000/9261, 6)

Exp term3 power over factorial = 4000/27783

Exp cumulative sum through3 := sum(1061/441, 4000/27783)

Exp cumulative sum through3 = 70843/27783

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

(20/21)^4

Exp ORIGINAL x power4 = 160000/194481

Exp term4 power over factorial := div(160000/194481, 24)

Exp term4 power over factorial = 20000/583443

Exp cumulative sum through4 := sum(70843/27783, 20000/583443)

Exp cumulative sum through4 = 1507703/583443

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

(20/21)^5

Exp ORIGINAL x power5 = 3200000/4084101

Exp term5 power over factorial := div(3200000/4084101, 120)

Exp term5 power over factorial = 80000/12252303

Exp cumulative sum through5 := sum(1507703/583443, 80000/12252303)

Exp cumulative sum through5 = 31741763/12252303

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

(20/21)^6

Exp ORIGINAL x power6 = 64000000/85766121

Exp term6 power over factorial := div(64000000/85766121, 720)

Exp term6 power over factorial = 800000/771895089

Exp cumulative sum through6 := sum(31741763/12252303, 800000/771895089)

Exp cumulative sum through6 = 2000531069/771895089

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

(20/21)^7

Exp ORIGINAL x power7 = 1280000000/1801088541

Exp term7 power over factorial := div(1280000000/1801088541, 5040)

Exp term7 power over factorial = 16000000/113468578083

Exp cumulative sum through7 := sum(2000531069/771895089, 16000000/113468578083)

Exp cumulative sum through7 = 294094067143/113468578083

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

(20/21)^8

Exp ORIGINAL x power8 = 25600000000/37822859361

Exp term8 power over factorial := div(25600000000/37822859361, 40320)

Exp term8 power over factorial = 40000000/2382840139743

Exp cumulative sum through8 := sum(294094067143/113468578083, 40000000/2382840139743)

Exp cumulative sum through8 = 6176015410003/2382840139743

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

(20/21)^9

Exp ORIGINAL x power9 = 512000000000/794280046581

Exp term9 power over factorial := div(512000000000/794280046581, 362880)

Exp term9 power over factorial = 800000000/450356786411427

Exp cumulative sum through9 := sum(6176015410003/2382840139743, 800000000/450356786411427)

Exp cumulative sum through9 = 1167267712490567/450356786411427

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

(20/21)^10

Exp ORIGINAL x power10 = 10240000000000/16679880978201

Exp term10 power over factorial := div(10240000000000/16679880978201, 3628800)

Exp term10 power over factorial = 1600000000/9457492514639967

Exp cumulative sum through10 := sum(1167267712490567/450356786411427, 1600000000/9457492514639967)

Exp cumulative sum through10 = 24512623562301907/9457492514639967

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

(20/21)^11

Exp ORIGINAL x power11 = 204800000000000/350277500542221

Exp term11 power over factorial := div(204800000000000/350277500542221, 39916800)

Exp term11 power over factorial = 32000000000/2184680770881832377

Exp cumulative sum through11 := sum(24512623562301907/9457492514639967, 32000000000/2184680770881832377)

Exp cumulative sum through11 = 5662416074891740517/2184680770881832377

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

(20/21)^12

Exp ORIGINAL x power12 = 4096000000000000/7355827511386641

Exp term12 power over factorial := div(4096000000000000/7355827511386641, 479001600)

Exp term12 power over factorial = 160000000000/137634888565555439751

Exp cumulative sum through12 := sum(5662416074891740517/2184680770881832377, 160000000000/137634888565555439751)

Exp cumulative sum through12 = 356732212878179652571/137634888565555439751

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

(20/21)^13

Exp ORIGINAL x power13 = 81920000000000000/154472377739119461

Exp term13 power over factorial := div(81920000000000000/154472377739119461, 6227020800)

Exp term13 power over factorial = 3200000000000/37574324578396635052023

Exp cumulative sum through13 := sum(356732212878179652571/137634888565555439751, 3200000000000/37574324578396635052023)

Exp cumulative sum through13 = 97387894118943045151883/37574324578396635052023

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

(20/21)^14

Exp ORIGINAL x power14 = 1638400000000000000/3243919932521508681

Exp term14 power over factorial := div(1638400000000000000/3243919932521508681, 87178291200)

Exp term14 power over factorial = 32000000000000/5523425713024305352647381

Exp cumulative sum through14 := sum(97387894118943045151883/37574324578396635052023, 32000000000000/5523425713024305352647381)

Exp cumulative sum through14 = 14316020435516627637326801/5523425713024305352647381

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

(20/21)^15

Exp ORIGINAL x power15 = 32768000000000000000/68122318582951682301

Exp term15 power over factorial := div(32768000000000000000/68122318582951682301, 1307674368000)

Exp term15 power over factorial = 128000000000000/347975819920531237216785003

Exp cumulative sum through15 := sum(14316020435516627637326801/5523425713024305352647381, 128000000000000/347975819920531237216785003)

Exp cumulative sum through15 = 901909287437675541151588463/347975819920531237216785003

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

(20/21)^16

Exp ORIGINAL x power16 = 655360000000000000000/1430568690241985328321

Exp term16 power over factorial := div(655360000000000000000/1430568690241985328321, 20922789888000)

Exp term16 power over factorial = 160000000000000/7307492218331155981552485063

Exp cumulative sum through16 := sum(901909287437675541151588463/347975819920531237216785003, 160000000000000/7307492218331155981552485063)

Exp cumulative sum through16 = 18940095036191346364183357723/7307492218331155981552485063

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

(20/21)^17

Exp ORIGINAL x power17 = 13107200000000000000000/30041942495081691894741

Exp term17 power over factorial := div(13107200000000000000000/30041942495081691894741, 355687428096000)

Exp term17 power over factorial = 3200000000000000/2608774721944222685414237167491

Exp cumulative sum through17 := sum(18940095036191346364183357723/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491)

Exp cumulative sum through17 = 6761613927920313852013458707111/2608774721944222685414237167491

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

(20/21)^18

Exp ORIGINAL x power18 = 262144000000000000000000/630880792396715529789561

Exp term18 power over factorial := div(262144000000000000000000/630880792396715529789561, 6402373705728000)

Exp term18 power over factorial = 32000000000000000/493058422447458087543290824655799

Exp cumulative sum through18 := sum(6761613927920313852013458707111/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799)

Exp cumulative sum through18 = 1277945032376939350030543695643979/493058422447458087543290824655799

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

(20/21)^19

Exp ORIGINAL x power19 = 5242880000000000000000000/13248496640331026125580781

Exp term19 power over factorial := div(5242880000000000000000000/13248496640331026125580781, 121645100408832000)

Exp term19 power over factorial = 640000000000000000/196730310556535776929773039037663801

Exp cumulative sum through19 := sum(1277945032376939350030543695643979/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801)

Exp cumulative sum through19 = 509900067918398801302186934561947621/196730310556535776929773039037663801

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

(20/21)^20

Exp ORIGINAL x power20 = 104857600000000000000000000/278218429446951548637196401

Exp term20 power over factorial := div(104857600000000000000000000/278218429446951548637196401, 2432902008176640000)

Exp term20 power over factorial = 640000000000000000/4131336521687251315525233819790939821

Exp cumulative sum through20 := sum(509900067918398801302186934561947621/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821)

Exp cumulative sum through20 = 10707901426286374827985925625800900041/4131336521687251315525233819790939821

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

(20/21)^21

Exp ORIGINAL x power21 = 2097152000000000000000000000/5842587018385982521381124421

Exp term21 power over factorial := div(2097152000000000000000000000/5842587018385982521381124421, 51090942171709440000)

Exp term21 power over factorial = 12800000000000000000/1821919406064077830146628114527804461061

Exp cumulative sum through21 := sum(10707901426286374827985925625800900041/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061)

Exp cumulative sum through21 = 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061

Exp term22 factorial step1 := mul(1, 1)

Exp term22 factorial step1 = 1

Exp term22 factorial step2 := mul(1, 2)

Exp term22 factorial step2 = 2

Exp term22 factorial step3 := mul(2, 3)

Exp term22 factorial step3 = 6

Exp term22 factorial step4 := mul(6, 4)

Exp term22 factorial step4 = 24

Exp term22 factorial step5 := mul(24, 5)

Exp term22 factorial step5 = 120

Exp term22 factorial step6 := mul(120, 6)

Exp term22 factorial step6 = 720

Exp term22 factorial step7 := mul(720, 7)

Exp term22 factorial step7 = 5040

Exp term22 factorial step8 := mul(5040, 8)

Exp term22 factorial step8 = 40320

Exp term22 factorial step9 := mul(40320, 9)

Exp term22 factorial step9 = 362880

Exp term22 factorial step10 := mul(362880, 10)

Exp term22 factorial step10 = 3628800

Exp term22 factorial step11 := mul(3628800, 11)

Exp term22 factorial step11 = 39916800

Exp term22 factorial step12 := mul(39916800, 12)

Exp term22 factorial step12 = 479001600

Exp term22 factorial step13 := mul(479001600, 13)

Exp term22 factorial step13 = 6227020800

Exp term22 factorial step14 := mul(6227020800, 14)

Exp term22 factorial step14 = 87178291200

Exp term22 factorial step15 := mul(87178291200, 15)

Exp term22 factorial step15 = 1307674368000

Exp term22 factorial step16 := mul(1307674368000, 16)

Exp term22 factorial step16 = 20922789888000

Exp term22 factorial step17 := mul(20922789888000, 17)

Exp term22 factorial step17 = 355687428096000

Exp term22 factorial step18 := mul(355687428096000, 18)

Exp term22 factorial step18 = 6402373705728000

Exp term22 factorial step19 := mul(6402373705728000, 19)

Exp term22 factorial step19 = 121645100408832000

Exp term22 factorial step20 := mul(121645100408832000, 20)

Exp term22 factorial step20 = 2432902008176640000

Exp term22 factorial step21 := mul(2432902008176640000, 21)

Exp term22 factorial step21 = 51090942171709440000

Exp term22 factorial step22 := mul(51090942171709440000, 22)

Exp term22 factorial step22 = 1124000727777607680000

(20/21)^22

Exp ORIGINAL x power22 = 41943040000000000000000000000/122694327386105632949003612841

Exp term22 power over factorial := div(41943040000000000000000000000/122694327386105632949003612841, 1124000727777607680000)

Exp term22 power over factorial = 128000000000000000000/420863382800801978763871094455922830505091

Exp cumulative sum through22 := sum(4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091)

Exp cumulative sum through22 = 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091

Exp term23 factorial step1 := mul(1, 1)

Exp term23 factorial step1 = 1

Exp term23 factorial step2 := mul(1, 2)

Exp term23 factorial step2 = 2

Exp term23 factorial step3 := mul(2, 3)

Exp term23 factorial step3 = 6

Exp term23 factorial step4 := mul(6, 4)

Exp term23 factorial step4 = 24

Exp term23 factorial step5 := mul(24, 5)

Exp term23 factorial step5 = 120

Exp term23 factorial step6 := mul(120, 6)

Exp term23 factorial step6 = 720

Exp term23 factorial step7 := mul(720, 7)

Exp term23 factorial step7 = 5040

Exp term23 factorial step8 := mul(5040, 8)

Exp term23 factorial step8 = 40320

Exp term23 factorial step9 := mul(40320, 9)

Exp term23 factorial step9 = 362880

Exp term23 factorial step10 := mul(362880, 10)

Exp term23 factorial step10 = 3628800

Exp term23 factorial step11 := mul(3628800, 11)

Exp term23 factorial step11 = 39916800

Exp term23 factorial step12 := mul(39916800, 12)

Exp term23 factorial step12 = 479001600

Exp term23 factorial step13 := mul(479001600, 13)

Exp term23 factorial step13 = 6227020800

Exp term23 factorial step14 := mul(6227020800, 14)

Exp term23 factorial step14 = 87178291200

Exp term23 factorial step15 := mul(87178291200, 15)

Exp term23 factorial step15 = 1307674368000

Exp term23 factorial step16 := mul(1307674368000, 16)

Exp term23 factorial step16 = 20922789888000

Exp term23 factorial step17 := mul(20922789888000, 17)

Exp term23 factorial step17 = 355687428096000

Exp term23 factorial step18 := mul(355687428096000, 18)

Exp term23 factorial step18 = 6402373705728000

Exp term23 factorial step19 := mul(6402373705728000, 19)

Exp term23 factorial step19 = 121645100408832000

Exp term23 factorial step20 := mul(121645100408832000, 20)

Exp term23 factorial step20 = 2432902008176640000

Exp term23 factorial step21 := mul(2432902008176640000, 21)

Exp term23 factorial step21 = 51090942171709440000

Exp term23 factorial step22 := mul(51090942171709440000, 22)

Exp term23 factorial step22 = 1124000727777607680000

Exp term23 factorial step23 := mul(1124000727777607680000, 23)

Exp term23 factorial step23 = 25852016738884976640000

(20/21)^23

Exp ORIGINAL x power23 = 838860800000000000000000000000/2576580875108218291929075869661

Exp term23 power over factorial := div(838860800000000000000000000000/2576580875108218291929075869661, 25852016738884976640000)

Exp term23 power over factorial = 2560000000000000000000/203277013892787355742949738622210727133958953

Exp cumulative sum through23 := sum(1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953)

Exp cumulative sum through23 = 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953

Exp term24 factorial step1 := mul(1, 1)

Exp term24 factorial step1 = 1

Exp term24 factorial step2 := mul(1, 2)

Exp term24 factorial step2 = 2

Exp term24 factorial step3 := mul(2, 3)

Exp term24 factorial step3 = 6

Exp term24 factorial step4 := mul(6, 4)

Exp term24 factorial step4 = 24

Exp term24 factorial step5 := mul(24, 5)

Exp term24 factorial step5 = 120

Exp term24 factorial step6 := mul(120, 6)

Exp term24 factorial step6 = 720

Exp term24 factorial step7 := mul(720, 7)

Exp term24 factorial step7 = 5040

Exp term24 factorial step8 := mul(5040, 8)

Exp term24 factorial step8 = 40320

Exp term24 factorial step9 := mul(40320, 9)

Exp term24 factorial step9 = 362880

Exp term24 factorial step10 := mul(362880, 10)

Exp term24 factorial step10 = 3628800

Exp term24 factorial step11 := mul(3628800, 11)

Exp term24 factorial step11 = 39916800

Exp term24 factorial step12 := mul(39916800, 12)

Exp term24 factorial step12 = 479001600

Exp term24 factorial step13 := mul(479001600, 13)

Exp term24 factorial step13 = 6227020800

Exp term24 factorial step14 := mul(6227020800, 14)

Exp term24 factorial step14 = 87178291200

Exp term24 factorial step15 := mul(87178291200, 15)

Exp term24 factorial step15 = 1307674368000

Exp term24 factorial step16 := mul(1307674368000, 16)

Exp term24 factorial step16 = 20922789888000

Exp term24 factorial step17 := mul(20922789888000, 17)

Exp term24 factorial step17 = 355687428096000

Exp term24 factorial step18 := mul(355687428096000, 18)

Exp term24 factorial step18 = 6402373705728000

Exp term24 factorial step19 := mul(6402373705728000, 19)

Exp term24 factorial step19 = 121645100408832000

Exp term24 factorial step20 := mul(121645100408832000, 20)

Exp term24 factorial step20 = 2432902008176640000

Exp term24 factorial step21 := mul(2432902008176640000, 21)

Exp term24 factorial step21 = 51090942171709440000

Exp term24 factorial step22 := mul(51090942171709440000, 22)

Exp term24 factorial step22 = 1124000727777607680000

Exp term24 factorial step23 := mul(1124000727777607680000, 23)

Exp term24 factorial step23 = 25852016738884976640000

Exp term24 factorial step24 := mul(25852016738884976640000, 24)

Exp term24 factorial step24 = 620448401733239439360000

(20/21)^24

Exp ORIGINAL x power24 = 16777216000000000000000000000000/54108198377272584130510593262881

Exp term24 power over factorial := div(16777216000000000000000000000000/54108198377272584130510593262881, 620448401733239439360000)

Exp term24 power over factorial = 6400000000000000000000/12806451875245603411805833533199275809439414039

Exp cumulative sum through24 := sum(526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039)

Exp cumulative sum through24 = 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039

source roles = [20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039]]

derived = [33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, No, finite sum is not a general infinite-value certificate]

Reverse checks

• S378 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S377 · premise_and_result_audit — Recompute Exp cumulative sum through24 with exact rational arithmetic, not rounded decimals.
• S376 · premise_and_result_audit — Recompute Exp term24 power over factorial with exact rational arithmetic, not rounded decimals.
• S375 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S374 · premise_and_result_audit — Recompute Exp term24 factorial step24 with exact rational arithmetic, not rounded decimals.
• S373 · premise_and_result_audit — Recompute Exp term24 factorial step23 with exact rational arithmetic, not rounded decimals.
• S372 · premise_and_result_audit — Recompute Exp term24 factorial step22 with exact rational arithmetic, not rounded decimals.
• S371 · premise_and_result_audit — Recompute Exp term24 factorial step21 with exact rational arithmetic, not rounded decimals.
• S370 · premise_and_result_audit — Recompute Exp term24 factorial step20 with exact rational arithmetic, not rounded decimals.
• S369 · premise_and_result_audit — Recompute Exp term24 factorial step19 with exact rational arithmetic, not rounded decimals.
• S368 · premise_and_result_audit — Recompute Exp term24 factorial step18 with exact rational arithmetic, not rounded decimals.
• S367 · premise_and_result_audit — Recompute Exp term24 factorial step17 with exact rational arithmetic, not rounded decimals.
• S366 · premise_and_result_audit — Recompute Exp term24 factorial step16 with exact rational arithmetic, not rounded decimals.
• S365 · premise_and_result_audit — Recompute Exp term24 factorial step15 with exact rational arithmetic, not rounded decimals.
• S364 · premise_and_result_audit — Recompute Exp term24 factorial step14 with exact rational arithmetic, not rounded decimals.
• S363 · premise_and_result_audit — Recompute Exp term24 factorial step13 with exact rational arithmetic, not rounded decimals.
• S362 · premise_and_result_audit — Recompute Exp term24 factorial step12 with exact rational arithmetic, not rounded decimals.
• S361 · premise_and_result_audit — Recompute Exp term24 factorial step11 with exact rational arithmetic, not rounded decimals.
• S360 · premise_and_result_audit — Recompute Exp term24 factorial step10 with exact rational arithmetic, not rounded decimals.
• S359 · premise_and_result_audit — Recompute Exp term24 factorial step9 with exact rational arithmetic, not rounded decimals.
• S358 · premise_and_result_audit — Recompute Exp term24 factorial step8 with exact rational arithmetic, not rounded decimals.
• S357 · premise_and_result_audit — Recompute Exp term24 factorial step7 with exact rational arithmetic, not rounded decimals.
• S356 · premise_and_result_audit — Recompute Exp term24 factorial step6 with exact rational arithmetic, not rounded decimals.
• S355 · premise_and_result_audit — Recompute Exp term24 factorial step5 with exact rational arithmetic, not rounded decimals.
• S354 · premise_and_result_audit — Recompute Exp term24 factorial step4 with exact rational arithmetic, not rounded decimals.
• S353 · premise_and_result_audit — Recompute Exp term24 factorial step3 with exact rational arithmetic, not rounded decimals.
• S352 · premise_and_result_audit — Recompute Exp term24 factorial step2 with exact rational arithmetic, not rounded decimals.
• S351 · premise_and_result_audit — Recompute Exp term24 factorial step1 with exact rational arithmetic, not rounded decimals.
• S350 · premise_and_result_audit — Recompute Exp cumulative sum through23 with exact rational arithmetic, not rounded decimals.
• S349 · premise_and_result_audit — Recompute Exp term23 power over factorial with exact rational arithmetic, not rounded decimals.
• S348 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S347 · premise_and_result_audit — Recompute Exp term23 factorial step23 with exact rational arithmetic, not rounded decimals.
• S346 · premise_and_result_audit — Recompute Exp term23 factorial step22 with exact rational arithmetic, not rounded decimals.
• S345 · premise_and_result_audit — Recompute Exp term23 factorial step21 with exact rational arithmetic, not rounded decimals.
• S344 · premise_and_result_audit — Recompute Exp term23 factorial step20 with exact rational arithmetic, not rounded decimals.
• S343 · premise_and_result_audit — Recompute Exp term23 factorial step19 with exact rational arithmetic, not rounded decimals.
• S342 · premise_and_result_audit — Recompute Exp term23 factorial step18 with exact rational arithmetic, not rounded decimals.
• S341 · premise_and_result_audit — Recompute Exp term23 factorial step17 with exact rational arithmetic, not rounded decimals.
• S340 · premise_and_result_audit — Recompute Exp term23 factorial step16 with exact rational arithmetic, not rounded decimals.
• S339 · premise_and_result_audit — Recompute Exp term23 factorial step15 with exact rational arithmetic, not rounded decimals.
• S338 · premise_and_result_audit — Recompute Exp term23 factorial step14 with exact rational arithmetic, not rounded decimals.
• S337 · premise_and_result_audit — Recompute Exp term23 factorial step13 with exact rational arithmetic, not rounded decimals.
• S336 · premise_and_result_audit — Recompute Exp term23 factorial step12 with exact rational arithmetic, not rounded decimals.
• S335 · premise_and_result_audit — Recompute Exp term23 factorial step11 with exact rational arithmetic, not rounded decimals.
• S334 · premise_and_result_audit — Recompute Exp term23 factorial step10 with exact rational arithmetic, not rounded decimals.
• S333 · premise_and_result_audit — Recompute Exp term23 factorial step9 with exact rational arithmetic, not rounded decimals.
• S332 · premise_and_result_audit — Recompute Exp term23 factorial step8 with exact rational arithmetic, not rounded decimals.
• S331 · premise_and_result_audit — Recompute Exp term23 factorial step7 with exact rational arithmetic, not rounded decimals.
• S330 · premise_and_result_audit — Recompute Exp term23 factorial step6 with exact rational arithmetic, not rounded decimals.
• S329 · premise_and_result_audit — Recompute Exp term23 factorial step5 with exact rational arithmetic, not rounded decimals.
• S328 · premise_and_result_audit — Recompute Exp term23 factorial step4 with exact rational arithmetic, not rounded decimals.
• S327 · premise_and_result_audit — Recompute Exp term23 factorial step3 with exact rational arithmetic, not rounded decimals.
• S326 · premise_and_result_audit — Recompute Exp term23 factorial step2 with exact rational arithmetic, not rounded decimals.
• S325 · premise_and_result_audit — Recompute Exp term23 factorial step1 with exact rational arithmetic, not rounded decimals.
• S324 · premise_and_result_audit — Recompute Exp cumulative sum through22 with exact rational arithmetic, not rounded decimals.
• S323 · premise_and_result_audit — Recompute Exp term22 power over factorial with exact rational arithmetic, not rounded decimals.
• S322 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S321 · premise_and_result_audit — Recompute Exp term22 factorial step22 with exact rational arithmetic, not rounded decimals.
• S320 · premise_and_result_audit — Recompute Exp term22 factorial step21 with exact rational arithmetic, not rounded decimals.
• S319 · premise_and_result_audit — Recompute Exp term22 factorial step20 with exact rational arithmetic, not rounded decimals.
• S318 · premise_and_result_audit — Recompute Exp term22 factorial step19 with exact rational arithmetic, not rounded decimals.
• S317 · premise_and_result_audit — Recompute Exp term22 factorial step18 with exact rational arithmetic, not rounded decimals.
• S316 · premise_and_result_audit — Recompute Exp term22 factorial step17 with exact rational arithmetic, not rounded decimals.
• S315 · premise_and_result_audit — Recompute Exp term22 factorial step16 with exact rational arithmetic, not rounded decimals.
• S314 · premise_and_result_audit — Recompute Exp term22 factorial step15 with exact rational arithmetic, not rounded decimals.
• S313 · premise_and_result_audit — Recompute Exp term22 factorial step14 with exact rational arithmetic, not rounded decimals.
• S312 · premise_and_result_audit — Recompute Exp term22 factorial step13 with exact rational arithmetic, not rounded decimals.
• S311 · premise_and_result_audit — Recompute Exp term22 factorial step12 with exact rational arithmetic, not rounded decimals.
• S310 · premise_and_result_audit — Recompute Exp term22 factorial step11 with exact rational arithmetic, not rounded decimals.
• S309 · premise_and_result_audit — Recompute Exp term22 factorial step10 with exact rational arithmetic, not rounded decimals.
• S308 · premise_and_result_audit — Recompute Exp term22 factorial step9 with exact rational arithmetic, not rounded decimals.
• S307 · premise_and_result_audit — Recompute Exp term22 factorial step8 with exact rational arithmetic, not rounded decimals.
• S306 · premise_and_result_audit — Recompute Exp term22 factorial step7 with exact rational arithmetic, not rounded decimals.
• S305 · premise_and_result_audit — Recompute Exp term22 factorial step6 with exact rational arithmetic, not rounded decimals.
• S304 · premise_and_result_audit — Recompute Exp term22 factorial step5 with exact rational arithmetic, not rounded decimals.
• S303 · premise_and_result_audit — Recompute Exp term22 factorial step4 with exact rational arithmetic, not rounded decimals.
• S302 · premise_and_result_audit — Recompute Exp term22 factorial step3 with exact rational arithmetic, not rounded decimals.
• S301 · premise_and_result_audit — Recompute Exp term22 factorial step2 with exact rational arithmetic, not rounded decimals.
• S300 · premise_and_result_audit — Recompute Exp term22 factorial step1 with exact rational arithmetic, not rounded decimals.
• 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

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

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.