第 24230 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":[5,6],"cutoff":9}。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":[5,6],"cutoff":9}。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]

(5/6)^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

(5/6)^1

Exp ORIGINAL x power1 = 5/6

Exp term1 power over factorial := div(5/6, 1)

Exp term1 power over factorial = 5/6

Exp cumulative sum through1 := sum(1, 5/6)

Exp cumulative sum through1 = 11/6

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

Exp term2 factorial step1 = 1

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

Exp term2 factorial step2 = 2

(5/6)^2

Exp ORIGINAL x power2 = 25/36

Exp term2 power over factorial := div(25/36, 2)

Exp term2 power over factorial = 25/72

Exp cumulative sum through2 := sum(11/6, 25/72)

Exp cumulative sum through2 = 157/72

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

(5/6)^3

Exp ORIGINAL x power3 = 125/216

Exp term3 power over factorial := div(125/216, 6)

Exp term3 power over factorial = 125/1296

Exp cumulative sum through3 := sum(157/72, 125/1296)

Exp cumulative sum through3 = 2951/1296

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

(5/6)^4

Exp ORIGINAL x power4 = 625/1296

Exp term4 power over factorial := div(625/1296, 24)

Exp term4 power over factorial = 625/31104

Exp cumulative sum through4 := sum(2951/1296, 625/31104)

Exp cumulative sum through4 = 71449/31104

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

(5/6)^5

Exp ORIGINAL x power5 = 3125/7776

Exp term5 power over factorial := div(3125/7776, 120)

Exp term5 power over factorial = 625/186624

Exp cumulative sum through5 := sum(71449/31104, 625/186624)

Exp cumulative sum through5 = 429319/186624

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

(5/6)^6

Exp ORIGINAL x power6 = 15625/46656

Exp term6 power over factorial := div(15625/46656, 720)

Exp term6 power over factorial = 3125/6718464

Exp cumulative sum through6 := sum(429319/186624, 3125/6718464)

Exp cumulative sum through6 = 15458609/6718464

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

(5/6)^7

Exp ORIGINAL x power7 = 78125/279936

Exp term7 power over factorial := div(78125/279936, 5040)

Exp term7 power over factorial = 15625/282175488

Exp cumulative sum through7 := sum(15458609/6718464, 15625/282175488)

Exp cumulative sum through7 = 649277203/282175488

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

(5/6)^8

Exp ORIGINAL x power8 = 390625/1679616

Exp term8 power over factorial := div(390625/1679616, 40320)

Exp term8 power over factorial = 78125/13544423424

Exp cumulative sum through8 := sum(649277203/282175488, 78125/13544423424)

Exp cumulative sum through8 = 31165383869/13544423424

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

(5/6)^9

Exp ORIGINAL x power9 = 1953125/10077696

Exp term9 power over factorial := div(1953125/10077696, 362880)

Exp term9 power over factorial = 390625/731398864896

Exp cumulative sum through9 := sum(31165383869/13544423424, 390625/731398864896)

Exp cumulative sum through9 = 1682931119551/731398864896

source roles = [5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896]]

derived = [1682931119551/731398864896, 否, finite sum is not a general infinite-value certificate]

逐步反向核對

• S078 · 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。
• 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

(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 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。