Question 24233: 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=8/9, cutoff=12, 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=8/9, cutoff=12, 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]

(8/9)^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

(8/9)^1

Exp ORIGINAL x power1 = 8/9

Exp term1 power over factorial := div(8/9, 1)

Exp term1 power over factorial = 8/9

Exp cumulative sum through1 := sum(1, 8/9)

Exp cumulative sum through1 = 17/9

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(8/9)^2

Exp ORIGINAL x power2 = 64/81

Exp term2 power over factorial := div(64/81, 2)

Exp term2 power over factorial = 32/81

Exp cumulative sum through2 := sum(17/9, 32/81)

Exp cumulative sum through2 = 185/81

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

(8/9)^3

Exp ORIGINAL x power3 = 512/729

Exp term3 power over factorial := div(512/729, 6)

Exp term3 power over factorial = 256/2187

Exp cumulative sum through3 := sum(185/81, 256/2187)

Exp cumulative sum through3 = 5251/2187

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

(8/9)^4

Exp ORIGINAL x power4 = 4096/6561

Exp term4 power over factorial := div(4096/6561, 24)

Exp term4 power over factorial = 512/19683

Exp cumulative sum through4 := sum(5251/2187, 512/19683)

Exp cumulative sum through4 = 47771/19683

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

(8/9)^5

Exp ORIGINAL x power5 = 32768/59049

Exp term5 power over factorial := div(32768/59049, 120)

Exp term5 power over factorial = 4096/885735

Exp cumulative sum through5 := sum(47771/19683, 4096/885735)

Exp cumulative sum through5 = 2153791/885735

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

(8/9)^6

Exp ORIGINAL x power6 = 262144/531441

Exp term6 power over factorial := div(262144/531441, 720)

Exp term6 power over factorial = 16384/23914845

Exp cumulative sum through6 := sum(2153791/885735, 16384/23914845)

Exp cumulative sum through6 = 58168741/23914845

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

(8/9)^7

Exp ORIGINAL x power7 = 2097152/4782969

Exp term7 power over factorial := div(2097152/4782969, 5040)

Exp term7 power over factorial = 131072/1506635235

Exp cumulative sum through7 := sum(58168741/23914845, 131072/1506635235)

Exp cumulative sum through7 = 732952351/301327047

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

(8/9)^8

Exp ORIGINAL x power8 = 16777216/43046721

Exp term8 power over factorial := div(16777216/43046721, 40320)

Exp term8 power over factorial = 131072/13559717115

Exp cumulative sum through8 := sum(732952351/301327047, 131072/13559717115)

Exp cumulative sum through8 = 32982986867/13559717115

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

(8/9)^9

Exp ORIGINAL x power9 = 134217728/387420489

Exp term9 power over factorial := div(134217728/387420489, 362880)

Exp term9 power over factorial = 1048576/1098337086315

Exp cumulative sum through9 := sum(32982986867/13559717115, 1048576/1098337086315)

Exp cumulative sum through9 = 2671622984803/1098337086315

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

(8/9)^10

Exp ORIGINAL x power10 = 1073741824/3486784401

Exp term10 power over factorial := div(1073741824/3486784401, 3628800)

Exp term10 power over factorial = 4194304/49425168884175

Exp cumulative sum through10 := sum(2671622984803/1098337086315, 4194304/49425168884175)

Exp cumulative sum through10 = 120223038510439/49425168884175

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

(8/9)^11

Exp ORIGINAL x power11 = 8589934592/31381059609

Exp term11 power over factorial := div(8589934592/31381059609, 39916800)

Exp term11 power over factorial = 33554432/4893091719533325

Exp cumulative sum through11 := sum(120223038510439/49425168884175, 33554432/4893091719533325)

Exp cumulative sum through11 = 11902080846087893/4893091719533325

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

(8/9)^12

Exp ORIGINAL x power12 = 68719476736/282429536481

Exp term12 power over factorial := div(68719476736/282429536481, 479001600)

Exp term12 power over factorial = 67108864/132113476427399775

Exp cumulative sum through12 := sum(11902080846087893/4893091719533325, 67108864/132113476427399775)

Exp cumulative sum through12 = 12854247316459279/5284539057095991

source roles = [8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991]]

derived = [12854247316459279/5284539057095991, No, finite sum is not a general infinite-value certificate]

Reverse checks

• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, No, finite_partial_sum_is_not_the_infinite_exponential_value)

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.