第 24233 題：實指數函數

原題與已知條件

E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[8,9],"cutoff":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[8,9],"cutoff":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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

• mko-real-exponential-function

逐步推導

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

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

source roles = [0]

derived = [1]

(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, 否, finite sum is not a general infinite-value certificate]

逐步反向核對

• S120 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S119 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through12，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Exp term12 power over factorial，不以小數近似替代。
• S117 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S116 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step12，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step11，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step10，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step9，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step8，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step7，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step6，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step5，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step4，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step3，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step2，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through11，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Exp term11 power over factorial，不以小數近似替代。
• S102 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S101 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step11，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step10，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step9，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step8，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step7，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step6，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step5，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step4，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step3，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step2，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step1，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through10，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Exp term10 power over factorial，不以小數近似替代。
• S088 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S087 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step10，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step9，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step8，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step7，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step6，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step5，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step4，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step3，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step2，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step1，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through9，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Exp term9 power over factorial，不以小數近似替代。
• S075 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S074 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step9，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step8，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step7，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step6，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step5，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step4，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step3，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step2，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step1，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through8，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Exp term8 power over factorial，不以小數近似替代。
• S063 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step8，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step7，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step6，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step5，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step4，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step3，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step2，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step1，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through7，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Exp term7 power over factorial，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step7，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step6，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step5，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step4，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step3，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step2，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step1，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through6，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Exp term6 power over factorial，不以小數近似替代。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step6，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step5，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step4，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step3，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step2，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through5，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Exp term5 power over factorial，不以小數近似替代。
• S033 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S032 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step5，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step4，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through4，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Exp term4 power over factorial，不以小數近似替代。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step4，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step3，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through3，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Exp term3 power over factorial，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Exp term2 power over factorial，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Exp term1 power over factorial，不以小數近似替代。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 Exp term1 factorial step1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Exp term0 power over factorial，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(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, 否, finite_partial_sum_is_not_the_infinite_exponential_value)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

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