第 24005 題：Taylor多項式

原題與已知條件

固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[5,10,15],"center":-8,"order":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={"derivatives":[5,10,15],"center":-8,"order":2}。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-taylor-polynomial

逐步推導

source roles = [taylor-coefficient-construction]

derived = [[center, order, shifted_coefficients, indexed_coefficients, certificate]]

source roles = [-8, 2, [5, 10, 15]]

derived = [[0, 1, 2], 3, 是]

source roles = [0]

derived = [1]

Taylor coefficient0 derivative over factorial := div(5, 1)

Taylor coefficient0 derivative over factorial = 5

Taylor k1 factorial step1 := mul(1, 1)

Taylor k1 factorial step1 = 1

Taylor coefficient1 derivative over factorial := div(10, 1)

Taylor coefficient1 derivative over factorial = 10

Taylor k2 factorial step1 := mul(1, 1)

Taylor k2 factorial step1 = 1

Taylor k2 factorial step2 := mul(1, 2)

Taylor k2 factorial step2 = 2

Taylor coefficient2 derivative over factorial := div(15, 2)

Taylor coefficient2 derivative over factorial = 15/2

source roles = [-8, 2, [5, 10, 15/2]]

derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate]

逐步反向核對

• S010 · 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。
• S009 · premise_and_result_audit — 以精確有理數重算 Taylor coefficient2 derivative over factorial，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Taylor k2 factorial step2，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Taylor k2 factorial step1，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Taylor coefficient1 derivative over factorial，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Taylor k1 factorial step1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Taylor coefficient0 derivative over factorial，不以小數近似替代。
• S003 · 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。
• S002 · 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。
• 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

(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)

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。