第 17945 題：絕對收斂

原題與已知條件

partial sums與absolute-series exact convergence certificate。操作=absseries-conditional-alternating；資料={"shift":20,"n":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-conditional-alternating；資料={"shift":20,"n":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-conditional-alternating;original question snapshot retained;source presentation interface resolved in this page

EXPLICIT ordered output fields=[shift, n, terms, series_converges, absolutely_converges, certificate]

alternating displayed term0 denominator := sum(0, 20)

alternating displayed term0 denominator = 20

alternating displayed term0 signed value := div(1, 20)

alternating displayed term0 signed value = 1/20

alternating displayed term1 denominator := sum(1, 20)

alternating displayed term1 denominator = 21

alternating displayed term1 signed value := div(-1, 21)

alternating displayed term1 signed value = -1/21

alternating displayed term2 denominator := sum(2, 20)

alternating displayed term2 denominator = 22

alternating displayed term2 signed value := div(1, 22)

alternating displayed term2 signed value = 1/22

alternating displayed term3 denominator := sum(3, 20)

alternating displayed term3 denominator = 23

alternating displayed term3 signed value := div(-1, 23)

alternating displayed term3 signed value = -1/23

alternating displayed term4 denominator := sum(4, 20)

alternating displayed term4 denominator = 24

alternating displayed term4 signed value := div(1, 24)

alternating displayed term4 signed value = 1/24

absolute series of declared alternating tail:DECLARED whole positive tail b_k=1/(k+20),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 series of declared alternating tail:same full absolute/positive series over denominators m=20,21,...

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

WHOLE declared terms=(-1)^k/(k+20),k starts0 positive;displayed first5 terms=[1/20, -1/21, 1/22, -1/23, 1/24]

b_k positive decreasing to0;even S_2m increases by b_2m-b_(2m+1)>=0;odd S_(2m+1) decreases;0<=S_2m<=S_(2m+1)<=b_0=1/20;gap b_2m tends0

alternating displayed exact partial sum := sum(1/20, -1/21, 1/22, -1/23, 1/24)

alternating displayed exact partial sum = 9781/212520

alternating next-term remainder denominator := sum(5, 20)

alternating next-term remainder denominator = 25

alternating next-term absolute remainder bound := div(1, 25)

alternating next-term absolute remainder bound = 1/25

alternating next signed term := mul(-1, 1/25)

alternating next signed term = -1/25

alternating next exact partial sum := sum(9781/212520, -1/25)

alternating next exact partial sum = 6401/1062600

displayed S_n=9781/212520;adjacent S_(n+1)=6401/1062600;n=5;positive shift=20

common limit L lies between these two endpoints;abs(L-S_n)<=1/25;original series converges=是;absolute harmonic series converges=否

逐步反向核對

• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 alternating next exact partial sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 alternating next signed term，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 alternating next-term absolute remainder bound，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 alternating next-term remainder denominator，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 alternating displayed exact partial sum，不以小數近似替代。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · premise_and_result_audit — 以精確有理數重算 alternating displayed term4 signed value，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 alternating displayed term4 denominator，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 alternating displayed term3 signed value，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 alternating displayed term3 denominator，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 alternating displayed term2 signed value，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 alternating displayed term2 denominator，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 alternating displayed term1 signed value，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 alternating displayed term1 denominator，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 alternating displayed term0 signed value，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 alternating displayed term0 denominator，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

(20, 5, [1/20, -1/21, 1/22, -1/23, 1/24], 是, 否, alternating_remainder_and_harmonic_absolute_divergence)

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。