第 17985 題：絕對收斂

原題與已知條件

partial sums與absolute-series exact convergence certificate。操作=absseries-term-zero-boundary；資料={"shift":10,"prefix_length":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — partial sums與absolute-series exact convergence certificate。操作=absseries-term-zero-boundary；資料={"shift":10,"prefix_length":5}。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-absolute-convergence-of-series

逐步推導

source operation=absseries-term-zero-boundary;original question snapshot retained;source presentation interface resolved in this page

EXPLICIT ordered output fields=[shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary]

absolute-convergence term-zero counterexample:DECLARED whole positive tail b_k=1/(k+10),k>=0;shift>0

b_k decreases to0:for ANY epsilon>0 choose N with N+shift>1/epsilon,then every k>=N has b_k<epsilon

absolute-convergence term-zero counterexample:same full absolute/positive series over denominators m=10,11,...

for every integer j>=0 large enough that2^j>=shift-1,the block m=2^j+1..2^(j+1) has2^j terms each>=1/2^(j+1);each block sum>=1/2;infinitely many disjoint blocks make partial sums unbounded

inspected finite prefix length=5;DECLARED harmonic tail shift=10;whole-tail term limit0 and divergent absolute sum proved separately

terms_tend_to_zero=是;term_zero_suffices_for_absolute_convergence=否;term_zero_not_absolute_convergence

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 — 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

(10, 5, 是, 否, term_zero_not_absolute_convergence)

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。