第 17701 題：無窮級數

原題與已知條件

固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={"a":[1,2],"r":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。

• G001 — 固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={"a":[1,2],"r":[1,2]}。請依 [a,r,infinite_sum,certificate] 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-infinite-series

逐步推導

signed geometric series one minus effective ratio := sub(1, 1/2)

signed geometric series one minus effective ratio = 1/2

signed geometric series exact infinite candidate sum := div(1/2, 1/2)

signed geometric series exact infinite candidate sum = 1

signed geometric series:original SIGNED a=1/2;SIGNED r=1/2;absolute ratio=1/2<1;effective terms=(1/2)*(1/2)^k,k=0,1,...

candidate sum=1;effective denominator1-ratio=1/2 is nonzero

signed geometric series:for ANY N>=1,the first N terms start at k0 with leading=1/2,ratio=1/2

(1-ratio)*S_N=(1/2)*(1-(1/2)^N);S_N=(1/2)*(1-(1/2)^N)/(1/2);candidate minus S_N=(1/2)*(1/2)^N/(1/2)

signed geometric series:|effective ratio|=1/2<1;remainder magnitude=abs(1/2)*abs(1/2)^N/abs(1/2)

geometric power tends to0;remainder tends to0;partial sums converge to the exact candidate

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 — 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 — 以精確有理數重算 signed geometric series exact infinite candidate sum，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 signed geometric series one minus effective ratio，不以小數近似替代。

結論

(1/2, 1/2, 1, absolute_ratio_below_one_symbolic_tail)

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。