第 17578 題：函數列逐點收斂

原題與已知條件

同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={"limit":[2,3],"coefficients":[3,-4],"shift":2,"prefix_length":3}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。

• G001 — 同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={"limit":[2,3],"coefficients":[3,-4],"shift":2,"prefix_length":3}。請依 [prefix_length,value_rows,sufficient,boundary] 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-pointwise-convergence-of-functions

逐步推導

same COMPLETE finite domain Fin(2);original aligned limits=[2, 3];signed coefficients=[3, -4];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite prefix point0 index0 denominator := sum(0, 2)

finite prefix point0 index0 denominator = 2

finite prefix point0 index0 signed correction := div(3, 2)

finite prefix point0 index0 signed correction = 3/2

finite prefix point0 index0 function value := sum(2, 3/2)

finite prefix point0 index0 function value = 7/2

finite prefix point0 index0 reverse signed error := sub(7/2, 2)

finite prefix point0 index0 reverse signed error = 3/2

finite prefix point0 index0:original point=0;n=0;limit=2;signed coefficient=3

f_n(x)=7/2;f_n(x)-limit=3/2;this error field remains SIGNED

finite prefix point0 index1 denominator := sum(1, 2)

finite prefix point0 index1 denominator = 3

finite prefix point0 index1 signed correction := div(3, 3)

finite prefix point0 index1 signed correction = 1

finite prefix point0 index1 function value := sum(2, 1)

finite prefix point0 index1 function value = 3

finite prefix point0 index1 reverse signed error := sub(3, 2)

finite prefix point0 index1 reverse signed error = 1

finite prefix point0 index1:original point=0;n=1;limit=2;signed coefficient=3

f_n(x)=3;f_n(x)-limit=1;this error field remains SIGNED

finite prefix point0 index2 denominator := sum(2, 2)

finite prefix point0 index2 denominator = 4

finite prefix point0 index2 signed correction := div(3, 4)

finite prefix point0 index2 signed correction = 3/4

finite prefix point0 index2 function value := sum(2, 3/4)

finite prefix point0 index2 function value = 11/4

finite prefix point0 index2 reverse signed error := sub(11/4, 2)

finite prefix point0 index2 reverse signed error = 3/4

finite prefix point0 index2:original point=0;n=2;limit=2;signed coefficient=3

f_n(x)=11/4;f_n(x)-limit=3/4;this error field remains SIGNED

finite prefix point1 index0 denominator := sum(0, 2)

finite prefix point1 index0 denominator = 2

finite prefix point1 index0 signed correction := div(-4, 2)

finite prefix point1 index0 signed correction = -2

finite prefix point1 index0 function value := sum(3, -2)

finite prefix point1 index0 function value = 1

finite prefix point1 index0 reverse signed error := sub(1, 3)

finite prefix point1 index0 reverse signed error = -2

finite prefix point1 index0:original point=1;n=0;limit=3;signed coefficient=-4

f_n(x)=1;f_n(x)-limit=-2;this error field remains SIGNED

finite prefix point1 index1 denominator := sum(1, 2)

finite prefix point1 index1 denominator = 3

finite prefix point1 index1 signed correction := div(-4, 3)

finite prefix point1 index1 signed correction = -4/3

finite prefix point1 index1 function value := sum(3, -4/3)

finite prefix point1 index1 function value = 5/3

finite prefix point1 index1 reverse signed error := sub(5/3, 3)

finite prefix point1 index1 reverse signed error = -4/3

finite prefix point1 index1:original point=1;n=1;limit=3;signed coefficient=-4

f_n(x)=5/3;f_n(x)-limit=-4/3;this error field remains SIGNED

finite prefix point1 index2 denominator := sum(2, 2)

finite prefix point1 index2 denominator = 4

finite prefix point1 index2 signed correction := div(-4, 4)

finite prefix point1 index2 signed correction = -1

finite prefix point1 index2 function value := sum(3, -1)

finite prefix point1 index2 function value = 2

finite prefix point1 index2 reverse signed error := sub(2, 3)

finite prefix point1 index2 reverse signed error = -1

finite prefix point1 index2:original point=1;n=2;limit=3;signed coefficient=-4

f_n(x)=2;f_n(x)-limit=-1;this error field remains SIGNED

finite observed length=3;complete domain-order rows=[[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]]

prefix-alone sufficiency=否;finite_prefix_not_pointwise_convergence_certificate

逐步反向核對

• S032 · 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。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index2 reverse signed error，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index2 function value，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index2 signed correction，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index2 denominator，不以小數近似替代。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index1 reverse signed error，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index1 function value，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index1 signed correction，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index1 denominator，不以小數近似替代。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index0 reverse signed error，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index0 function value，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index0 signed correction，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 finite prefix point1 index0 denominator，不以小數近似替代。
• S016 · 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。
• S015 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index2 reverse signed error，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index2 function value，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index2 signed correction，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index2 denominator，不以小數近似替代。
• S011 · 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。
• S010 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index1 reverse signed error，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index1 function value，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index1 signed correction，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index1 denominator，不以小數近似替代。
• 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 finite prefix point0 index0 reverse signed error，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index0 function value，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index0 signed correction，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 finite prefix point0 index0 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

(3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], 否, finite_prefix_not_pointwise_convergence_certificate)

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。