第 23633 題：瑕積分

原題與已知條件

題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={"scale":8,"exponent":5,"lower":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={"scale":8,"exponent":5,"lower":3}。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-improper-integral

逐步推導

source operation=improper-one-sided-tail;original snapshot and mathematics retained

explicit ordered source output fields=[integrand, interval, integral, converges, certificate]

Improper ORIGINAL positive tail lower positive power step1 := mul(1, 3)

Improper ORIGINAL positive tail lower positive power step1 = 3

Improper ORIGINAL positive tail lower positive power step2 := mul(3, 3)

Improper ORIGINAL positive tail lower positive power step2 = 9

Improper ORIGINAL positive tail lower positive power step3 := mul(9, 3)

Improper ORIGINAL positive tail lower positive power step3 = 27

Improper ORIGINAL positive tail lower positive power step4 := mul(27, 3)

Improper ORIGINAL positive tail lower positive power step4 = 81

Improper ORIGINAL positive tail exponent-minus-one times lower-power := mul(4, 81)

Improper ORIGINAL positive tail exponent-minus-one times lower-power = 324

Improper ORIGINAL positive tail exact antiderivative limit value := div(8, 324)

Improper ORIGINAL positive tail exact antiderivative limit value = 2/81

Improper ORIGINAL positive tail:ORIGINAL scale=8,exponent=5>1,lower=3>=1;integrand scale/x^exponent

for EVERY B>=lower,integral_lower^B=scale/(exponent-1)*(lower^(1-exponent)-B^(1-exponent));B^(1-exponent) tends0,so limit=2/81

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 — 以精確有理數重算 Improper ORIGINAL positive tail exact antiderivative limit value，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail exponent-minus-one times lower-power，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step4，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step3，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step2，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Improper ORIGINAL positive tail lower positive power step1，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

(f(x)=8/x^5, [3,infinity), 2/81, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)

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。