第 26945 題：級數比較判別法

原題與已知條件

非負項、eventual forall比較方向與reference series狀態責任。操作=series-eventual-tail；資料={"prefix_a":[20,21,22,23],"prefix_b":[1,2,3,4],"a_scale":20,"b_scale":21,"ratio":[5,12],"cutoff":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 非負項、eventual forall比較方向與reference series狀態責任。操作=series-eventual-tail；資料={"prefix_a":[20,21,22,23],"prefix_b":[1,2,3,4],"a_scale":20,"b_scale":21,"ratio":[5,12],"cutoff":4}。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-series-comparison-test

逐步推導

source roles = [series-eventual-tail]

derived = [[cutoff, prefix_a_sum, prefix_b_sum, tail_a_sum, tail_b_sum, total_a, convergent, certificate]]

source roles = [20, 21, 5/12]

derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, 是]

Series prefixA ORIGINAL running sum0 := sum(0, 20)

Series prefixA ORIGINAL running sum0 = 20

Series prefixA ORIGINAL running sum1 := sum(20, 21)

Series prefixA ORIGINAL running sum1 = 41

Series prefixA ORIGINAL running sum2 := sum(41, 22)

Series prefixA ORIGINAL running sum2 = 63

Series prefixA ORIGINAL running sum3 := sum(63, 23)

Series prefixA ORIGINAL running sum3 = 86

Series prefixB ORIGINAL running sum0 := sum(0, 1)

Series prefixB ORIGINAL running sum0 = 1

Series prefixB ORIGINAL running sum1 := sum(1, 2)

Series prefixB ORIGINAL running sum1 = 3

Series prefixB ORIGINAL running sum2 := sum(3, 3)

Series prefixB ORIGINAL running sum2 = 6

Series prefixB ORIGINAL running sum3 := sum(6, 4)

Series prefixB ORIGINAL running sum3 = 10

source roles = [4, [20, 21, 22, 23], [1, 2, 3, 4]]

derived = [86, 10, tail exponent n-cutoff starts at0]

Series ORIGINAL one minus ratio := sub(1, 5/12)

Series ORIGINAL one minus ratio = 7/12

Series ORIGINAL geometric tailA := div(20, 7/12)

Series ORIGINAL geometric tailA = 240/7

Series ORIGINAL geometric tailB := div(21, 7/12)

Series ORIGINAL geometric tailB = 36

source roles = [20, 21, 5/12, 7/12]

derived = [240/7, 36, 是]

Series ORIGINAL totalA prefix plus tail := sum(86, 240/7)

Series ORIGINAL totalA prefix plus tail = 842/7

source roles = [4, 86, 10, 240/7, 36]

derived = [842/7, 是]

逐步反向核對

• S017 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。
• S016 · premise_and_result_audit — 以精確有理數重算 Series ORIGINAL totalA prefix plus tail，不以小數近似替代。
• S015 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。
• S014 · premise_and_result_audit — 以精確有理數重算 Series ORIGINAL geometric tailB，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Series ORIGINAL geometric tailA，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Series ORIGINAL one minus ratio，不以小數近似替代。
• S011 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。
• S010 · premise_and_result_audit — 以精確有理數重算 Series prefixB ORIGINAL running sum3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Series prefixB ORIGINAL running sum2，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Series prefixB ORIGINAL running sum1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Series prefixB ORIGINAL running sum0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Series prefixA ORIGINAL running sum3，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Series prefixA ORIGINAL running sum2，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Series prefixA ORIGINAL running sum1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Series prefixA ORIGINAL running sum0，不以小數近似替代。
• S002 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。
• S001 · premise_and_result_audit — 反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。

結論

(4, 86, 10, 240/7, 36, 842/7, 是, finite_prefix_does_not_change_tail_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。