第 24257 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-infinite-radius-certificate；資料={"epsilon_denominator":8,"start_index":8}。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-infinite-radius-certificate；資料={"epsilon_denominator":8,"start_index":8}。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-infinite-radius-certificate]

derived = [[epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate]]

Exp tail coefficient ratio bound := div(1, 9)

Exp tail coefficient ratio bound = 1/9

Exp declared positive epsilon := div(1, 8)

Exp declared positive epsilon = 1/8

source roles = [8, 8, 1/9, 1/8]

derived = [EVERY n>=N:1/(n+1)<=1/(N+1)<1/d, for ANY epsilon>0 choose integer N with N+1>1/epsilon, coefficient ratio limit0,radius infinity]

source roles = [[8, 8], a_n=1/n! for EVERY n>=0]

derived = [coefficient ratio=1/(n+1), for EVERY fixed real z,term absolute ratio=|z|/(n+1)->0, absolute convergence and real-valued E(z)]

逐步反向核對

• S005 · 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。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 Exp declared positive epsilon，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Exp tail coefficient ratio bound，不以小數近似替代。
• 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, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)

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。