第 23878 題：冪級數收斂半徑

原題與已知條件

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":3,"prefix_length":1,"first_ratio":[1,2],"second_ratio":[1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":3,"prefix_length":1,"first_ratio":[1,2],"second_ratio":[1,3]}。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-power-series-radius-of-convergence

逐步推導

source operation=radius-zero-infinity-prefix-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate]

Radius ORIGINAL first complete-tail reciprocal := div(1, 1/2)

Radius ORIGINAL first complete-tail reciprocal = 2

Radius ORIGINAL second complete-tail reciprocal := div(1, 1/3)

Radius ORIGINAL second complete-tail reciprocal = 3

ORIGINAL prefix_length parameter N=1;positive scale=3;distinct positive tail ratios=[1/2, 1/3]

COMMON zero coefficients at ALL n0..N=[0, 0];there are N+1=2 zeros;for n>N laws are scale*q^(n-N-1),first tail term scale;radii=[2, 3] DIFFER

逐步反向核對

• 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 Radius ORIGINAL second complete-tail reciprocal，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Radius ORIGINAL first complete-tail reciprocal，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

(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

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。