第 24231 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":[6,7],"cutoff":10}。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":[6,7],"cutoff":10}。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]

(6/7)^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

(6/7)^1

Exp ORIGINAL x power1 = 6/7

Exp term1 power over factorial := div(6/7, 1)

Exp term1 power over factorial = 6/7

Exp cumulative sum through1 := sum(1, 6/7)

Exp cumulative sum through1 = 13/7

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(6/7)^2

Exp ORIGINAL x power2 = 36/49

Exp term2 power over factorial := div(36/49, 2)

Exp term2 power over factorial = 18/49

Exp cumulative sum through2 := sum(13/7, 18/49)

Exp cumulative sum through2 = 109/49

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

(6/7)^3

Exp ORIGINAL x power3 = 216/343

Exp term3 power over factorial := div(216/343, 6)

Exp term3 power over factorial = 36/343

Exp cumulative sum through3 := sum(109/49, 36/343)

Exp cumulative sum through3 = 799/343

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

(6/7)^4

Exp ORIGINAL x power4 = 1296/2401

Exp term4 power over factorial := div(1296/2401, 24)

Exp term4 power over factorial = 54/2401

Exp cumulative sum through4 := sum(799/343, 54/2401)

Exp cumulative sum through4 = 5647/2401

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

(6/7)^5

Exp ORIGINAL x power5 = 7776/16807

Exp term5 power over factorial := div(7776/16807, 120)

Exp term5 power over factorial = 324/84035

Exp cumulative sum through5 := sum(5647/2401, 324/84035)

Exp cumulative sum through5 = 197969/84035

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

(6/7)^6

Exp ORIGINAL x power6 = 46656/117649

Exp term6 power over factorial := div(46656/117649, 720)

Exp term6 power over factorial = 324/588245

Exp cumulative sum through6 := sum(197969/84035, 324/588245)

Exp cumulative sum through6 = 1386107/588245

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

(6/7)^7

Exp ORIGINAL x power7 = 279936/823543

Exp term7 power over factorial := div(279936/823543, 5040)

Exp term7 power over factorial = 1944/28824005

Exp cumulative sum through7 := sum(1386107/588245, 1944/28824005)

Exp cumulative sum through7 = 67921187/28824005

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

(6/7)^8

Exp ORIGINAL x power8 = 1679616/5764801

Exp term8 power over factorial := div(1679616/5764801, 40320)

Exp term8 power over factorial = 1458/201768035

Exp cumulative sum through8 := sum(67921187/28824005, 1458/201768035)

Exp cumulative sum through8 = 475449767/201768035

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

(6/7)^9

Exp ORIGINAL x power9 = 10077696/40353607

Exp term9 power over factorial := div(10077696/40353607, 362880)

Exp term9 power over factorial = 972/1412376245

Exp cumulative sum through9 := sum(475449767/201768035, 972/1412376245)

Exp cumulative sum through9 = 3328149341/1412376245

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

(6/7)^10

Exp ORIGINAL x power10 = 60466176/282475249

Exp term10 power over factorial := div(60466176/282475249, 3628800)

Exp term10 power over factorial = 2916/49433168575

Exp cumulative sum through10 := sum(3328149341/1412376245, 2916/49433168575)

Exp cumulative sum through10 = 116485229851/49433168575

source roles = [6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575]]

derived = [116485229851/49433168575, 否, finite sum is not a general infinite-value certificate]

逐步反向核對

• S091 · 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。
• 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

(6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, 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。