第 17757 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-telescoping-sum；資料={"c":7,"shift":2,"n":7}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。

• G001 — 固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={"c":7,"shift":2,"n":7}。請依 [c,shift,n,partial_sum,remainder,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

telescoping final remainder denominator := sum(2, 7)

telescoping final remainder denominator = 9

telescoping first endpoint numeric value := div(7, 2)

telescoping first endpoint numeric value = 7/2

telescoping signed remainder numeric value := div(7, 9)

telescoping signed remainder numeric value = 7/9

telescoping reduced partial sum := sub(7/2, 7/9)

telescoping reduced partial sum = 49/18

telescoping partial plus signed remainder := sum(49/18, 7/9)

telescoping partial plus signed remainder = 7/2

original c=7;shift=2;n=7;terms c/(shift+k)-c/(shift+k+1),k0..n-1

ALL interior terms cancel;S_n=c/shift-c/(shift+n)=49/18;signed remainder=7/9;reverse sum=7/2

same source formatting c=7;source remainder ratio text=7/9;source infinite-sum ratio text=7/2

computed partial sum REDUCED=49/18;remainder/infinite fields retain DIRECT UNREDUCED source text;numeric equivalents=[7/9, 7/2]

DECLARED whole telescoping tail;remainder at general n is(7)/(n+2)

signed remainder tends to0;full partial-sum limit=c/shift under the given index convention

逐步反向核對

• S008 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 telescoping partial plus signed remainder，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 telescoping reduced partial sum，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 telescoping signed remainder numeric value，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 telescoping first endpoint numeric value，不以小數近似替代。
• S001 · premise_and_result_audit — 以精確有理數重算 telescoping final remainder denominator，不以小數近似替代。

結論

(7, 2, 7, 49/18, 7/9, 7/2, exact_telescoping_remainder)

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。