第 26420 題：Taylor餘項定理

原題與已知條件

center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={"mode":"undersized_M","coefficients":[4,7,10,13,16,2,21],"center":0,"x":[1,2],"order":5,"M":15119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={"mode":"undersized_M","coefficients":[4,7,10,13,16,2,21],"center":0,"x":[1,2],"order":5,"M":15119}。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-remainder-theorem

逐步推導

source roles = [taylor-bound-boundary]

derived = [[mode, remainder, derivative_supremum, M, proposed_error_bound, actual_error_inequality, qualified_bound_certificate, general_theorem_or_analyticity_claim, boundary]]

Taylor ORIGINAL x minus declared center := sub(1/2, 0)

Taylor ORIGINAL x minus declared center = 1/2

(0)^0

Taylor ORIGINAL derivative0 center power0 term0 = 1

Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)

Taylor ORIGINAL derivative0 falling factor times power term0 = 1

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(4, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 4

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 4)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 4

(0)^1

Taylor ORIGINAL derivative0 center power1 term1 = 0

Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term1 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(7, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 4

(0)^2

Taylor ORIGINAL derivative0 center power2 term2 = 0

Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term2 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(10, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 4

(0)^3

Taylor ORIGINAL derivative0 center power3 term3 = 0

Taylor ORIGINAL derivative0 falling factor times power term3 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term3 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(13, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 4

(0)^4

Taylor ORIGINAL derivative0 center power4 term4 = 0

Taylor ORIGINAL derivative0 falling factor times power term4 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term4 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 := mul(16, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term4 = 4

(0)^5

Taylor ORIGINAL derivative0 center power5 term5 = 0

Taylor ORIGINAL derivative0 falling factor times power term5 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term5 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 := mul(2, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term5 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term5 = 4

(0)^6

Taylor ORIGINAL derivative0 center power6 term6 = 0

Taylor ORIGINAL derivative0 falling factor times power term6 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term6 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term6 := sum(4, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term6 = 4

source roles = [0, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[0, 4, 1, 1, 4], [1, 7, 1, 0, 0], [2, 10, 1, 0, 0], [3, 13, 1, 0, 0], [4, 16, 1, 0, 0], [5, 2, 1, 0, 0], [6, 21, 1, 0, 0]], 4]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(4, 1)

Taylor ORIGINAL derivative times displacement0 = 4

Taylor ORIGINAL term0 divide factorial := div(4, 1)

Taylor ORIGINAL term0 divide factorial = 4

(0)^0

Taylor ORIGINAL derivative1 center power0 term1 = 1

Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)

Taylor ORIGINAL derivative1 falling factor times power term1 = 1

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(7, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 7

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 7)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 7

(0)^1

Taylor ORIGINAL derivative1 center power1 term2 = 0

Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)

Taylor ORIGINAL derivative1 falling factor times power term2 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(10, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(7, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 7

(0)^2

Taylor ORIGINAL derivative1 center power2 term3 = 0

Taylor ORIGINAL derivative1 falling factor times power term3 := mul(3, 0)

Taylor ORIGINAL derivative1 falling factor times power term3 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(13, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(7, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 7

(0)^3

Taylor ORIGINAL derivative1 center power3 term4 = 0

Taylor ORIGINAL derivative1 falling factor times power term4 := mul(4, 0)

Taylor ORIGINAL derivative1 falling factor times power term4 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 := mul(16, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(7, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term4 = 7

(0)^4

Taylor ORIGINAL derivative1 center power4 term5 = 0

Taylor ORIGINAL derivative1 falling factor times power term5 := mul(5, 0)

Taylor ORIGINAL derivative1 falling factor times power term5 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 := mul(2, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term5 := sum(7, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term5 = 7

(0)^5

Taylor ORIGINAL derivative1 center power5 term6 = 0

Taylor ORIGINAL derivative1 falling factor times power term6 := mul(6, 0)

Taylor ORIGINAL derivative1 falling factor times power term6 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term6 := sum(7, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term6 = 7

source roles = [1, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[1, 7, 1, 1, 7], [2, 10, 2, 0, 0], [3, 13, 3, 0, 0], [4, 16, 4, 0, 0], [5, 2, 5, 0, 0], [6, 21, 6, 0, 0]], 7]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(7, 1/2)

Taylor ORIGINAL derivative times displacement1 = 7/2

Taylor ORIGINAL term1 divide factorial := div(7/2, 1)

Taylor ORIGINAL term1 divide factorial = 7/2

(0)^0

Taylor ORIGINAL derivative2 center power0 term2 = 1

Taylor ORIGINAL derivative2 falling factor times power term2 := mul(2, 1)

Taylor ORIGINAL derivative2 falling factor times power term2 = 2

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(10, 2)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 20

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 20)

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 20

(0)^1

Taylor ORIGINAL derivative2 center power1 term3 = 0

Taylor ORIGINAL derivative2 falling factor times power term3 := mul(6, 0)

Taylor ORIGINAL derivative2 falling factor times power term3 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(13, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(20, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 20

(0)^2

Taylor ORIGINAL derivative2 center power2 term4 = 0

Taylor ORIGINAL derivative2 falling factor times power term4 := mul(12, 0)

Taylor ORIGINAL derivative2 falling factor times power term4 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 := mul(16, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(20, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term4 = 20

(0)^3

Taylor ORIGINAL derivative2 center power3 term5 = 0

Taylor ORIGINAL derivative2 falling factor times power term5 := mul(20, 0)

Taylor ORIGINAL derivative2 falling factor times power term5 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 := mul(2, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term5 := sum(20, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term5 = 20

(0)^4

Taylor ORIGINAL derivative2 center power4 term6 = 0

Taylor ORIGINAL derivative2 falling factor times power term6 := mul(30, 0)

Taylor ORIGINAL derivative2 falling factor times power term6 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term6 := sum(20, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term6 = 20

source roles = [2, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[2, 10, 2, 1, 20], [3, 13, 6, 0, 0], [4, 16, 12, 0, 0], [5, 2, 20, 0, 0], [6, 21, 30, 0, 0]], 20]

(1/2)^2

Taylor ORIGINAL displacement power2 = 1/4

Taylor ORIGINAL derivative times displacement2 := mul(20, 1/4)

Taylor ORIGINAL derivative times displacement2 = 5

Taylor ORIGINAL term2 divide factorial := div(5, 2)

Taylor ORIGINAL term2 divide factorial = 5/2

(0)^0

Taylor ORIGINAL derivative3 center power0 term3 = 1

Taylor ORIGINAL derivative3 falling factor times power term3 := mul(6, 1)

Taylor ORIGINAL derivative3 falling factor times power term3 = 6

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 := mul(13, 6)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 78

Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 78)

Taylor ORIGINAL derivative3 ascending derivative sum term3 = 78

(0)^1

Taylor ORIGINAL derivative3 center power1 term4 = 0

Taylor ORIGINAL derivative3 falling factor times power term4 := mul(24, 0)

Taylor ORIGINAL derivative3 falling factor times power term4 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 := mul(16, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(78, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term4 = 78

(0)^2

Taylor ORIGINAL derivative3 center power2 term5 = 0

Taylor ORIGINAL derivative3 falling factor times power term5 := mul(60, 0)

Taylor ORIGINAL derivative3 falling factor times power term5 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 := mul(2, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term5 := sum(78, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term5 = 78

(0)^3

Taylor ORIGINAL derivative3 center power3 term6 = 0

Taylor ORIGINAL derivative3 falling factor times power term6 := mul(120, 0)

Taylor ORIGINAL derivative3 falling factor times power term6 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term6 := sum(78, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term6 = 78

source roles = [3, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[3, 13, 6, 1, 78], [4, 16, 24, 0, 0], [5, 2, 60, 0, 0], [6, 21, 120, 0, 0]], 78]

(1/2)^3

Taylor ORIGINAL displacement power3 = 1/8

Taylor ORIGINAL derivative times displacement3 := mul(78, 1/8)

Taylor ORIGINAL derivative times displacement3 = 39/4

Taylor ORIGINAL term3 divide factorial := div(39/4, 6)

Taylor ORIGINAL term3 divide factorial = 13/8

(0)^0

Taylor ORIGINAL derivative4 center power0 term4 = 1

Taylor ORIGINAL derivative4 falling factor times power term4 := mul(24, 1)

Taylor ORIGINAL derivative4 falling factor times power term4 = 24

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 := mul(16, 24)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 = 384

Taylor ORIGINAL derivative4 ascending derivative sum term4 := sum(0, 384)

Taylor ORIGINAL derivative4 ascending derivative sum term4 = 384

(0)^1

Taylor ORIGINAL derivative4 center power1 term5 = 0

Taylor ORIGINAL derivative4 falling factor times power term5 := mul(120, 0)

Taylor ORIGINAL derivative4 falling factor times power term5 = 0

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 := mul(2, 0)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative4 ascending derivative sum term5 := sum(384, 0)

Taylor ORIGINAL derivative4 ascending derivative sum term5 = 384

(0)^2

Taylor ORIGINAL derivative4 center power2 term6 = 0

Taylor ORIGINAL derivative4 falling factor times power term6 := mul(360, 0)

Taylor ORIGINAL derivative4 falling factor times power term6 = 0

Taylor ORIGINAL derivative4 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative4 ascending derivative sum term6 := sum(384, 0)

Taylor ORIGINAL derivative4 ascending derivative sum term6 = 384

source roles = [4, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[4, 16, 24, 1, 384], [5, 2, 120, 0, 0], [6, 21, 360, 0, 0]], 384]

(1/2)^4

Taylor ORIGINAL displacement power4 = 1/16

Taylor ORIGINAL derivative times displacement4 := mul(384, 1/16)

Taylor ORIGINAL derivative times displacement4 = 24

Taylor ORIGINAL term4 divide factorial := div(24, 24)

Taylor ORIGINAL term4 divide factorial = 1

(0)^0

Taylor ORIGINAL derivative5 center power0 term5 = 1

Taylor ORIGINAL derivative5 falling factor times power term5 := mul(120, 1)

Taylor ORIGINAL derivative5 falling factor times power term5 = 120

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 := mul(2, 120)

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 = 240

Taylor ORIGINAL derivative5 ascending derivative sum term5 := sum(0, 240)

Taylor ORIGINAL derivative5 ascending derivative sum term5 = 240

(0)^1

Taylor ORIGINAL derivative5 center power1 term6 = 0

Taylor ORIGINAL derivative5 falling factor times power term6 := mul(720, 0)

Taylor ORIGINAL derivative5 falling factor times power term6 = 0

Taylor ORIGINAL derivative5 ORIGINAL coefficient term6 := mul(21, 0)

Taylor ORIGINAL derivative5 ORIGINAL coefficient term6 = 0

Taylor ORIGINAL derivative5 ascending derivative sum term6 := sum(240, 0)

Taylor ORIGINAL derivative5 ascending derivative sum term6 = 240

source roles = [5, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[5, 2, 120, 1, 240], [6, 21, 720, 0, 0]], 240]

(1/2)^5

Taylor ORIGINAL displacement power5 = 1/32

Taylor ORIGINAL derivative times displacement5 := mul(240, 1/32)

Taylor ORIGINAL derivative times displacement5 = 15/2

Taylor ORIGINAL term5 divide factorial := div(15/2, 120)

Taylor ORIGINAL term5 divide factorial = 1/16

Taylor ORIGINAL partial sum0 := sum(0, 4)

Taylor ORIGINAL partial sum0 = 4

Taylor ORIGINAL partial sum1 := sum(4, 7/2)

Taylor ORIGINAL partial sum1 = 15/2

Taylor ORIGINAL partial sum2 := sum(15/2, 5/2)

Taylor ORIGINAL partial sum2 = 10

Taylor ORIGINAL partial sum3 := sum(10, 13/8)

Taylor ORIGINAL partial sum3 = 93/8

Taylor ORIGINAL partial sum4 := sum(93/8, 1)

Taylor ORIGINAL partial sum4 = 101/8

Taylor ORIGINAL partial sum5 := sum(101/8, 1/16)

Taylor ORIGINAL partial sum5 = 203/16

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(4, 1)

Taylor ORIGINAL full function at x coefficient term0 = 4

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 4)

Taylor ORIGINAL full function at x ascending partial sum0 = 4

(1/2)^1

Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2

Taylor ORIGINAL full function at x coefficient term1 := mul(7, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 7/2

Taylor ORIGINAL full function at x ascending partial sum1 := sum(4, 7/2)

Taylor ORIGINAL full function at x ascending partial sum1 = 15/2

(1/2)^2

Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4

Taylor ORIGINAL full function at x coefficient term2 := mul(10, 1/4)

Taylor ORIGINAL full function at x coefficient term2 = 5/2

Taylor ORIGINAL full function at x ascending partial sum2 := sum(15/2, 5/2)

Taylor ORIGINAL full function at x ascending partial sum2 = 10

(1/2)^3

Taylor ORIGINAL full function at x ORIGINAL argument power3 = 1/8

Taylor ORIGINAL full function at x coefficient term3 := mul(13, 1/8)

Taylor ORIGINAL full function at x coefficient term3 = 13/8

Taylor ORIGINAL full function at x ascending partial sum3 := sum(10, 13/8)

Taylor ORIGINAL full function at x ascending partial sum3 = 93/8

(1/2)^4

Taylor ORIGINAL full function at x ORIGINAL argument power4 = 1/16

Taylor ORIGINAL full function at x coefficient term4 := mul(16, 1/16)

Taylor ORIGINAL full function at x coefficient term4 = 1

Taylor ORIGINAL full function at x ascending partial sum4 := sum(93/8, 1)

Taylor ORIGINAL full function at x ascending partial sum4 = 101/8

(1/2)^5

Taylor ORIGINAL full function at x ORIGINAL argument power5 = 1/32

Taylor ORIGINAL full function at x coefficient term5 := mul(2, 1/32)

Taylor ORIGINAL full function at x coefficient term5 = 1/16

Taylor ORIGINAL full function at x ascending partial sum5 := sum(101/8, 1/16)

Taylor ORIGINAL full function at x ascending partial sum5 = 203/16

(1/2)^6

Taylor ORIGINAL full function at x ORIGINAL argument power6 = 1/64

Taylor ORIGINAL full function at x coefficient term6 := mul(21, 1/64)

Taylor ORIGINAL full function at x coefficient term6 = 21/64

Taylor ORIGINAL full function at x ascending partial sum6 := sum(203/16, 21/64)

Taylor ORIGINAL full function at x ascending partial sum6 = 833/64

Taylor ORIGINAL f(x) minus Taylor value := sub(833/64, 203/16)

Taylor ORIGINAL f(x) minus Taylor value = 21/64

source roles = [[4, 7, 10, 13, 16, 2, 21], 0, 1/2, 5, [4, 7, 20, 78, 384, 240], [4, 7/2, 5/2, 13/8, 1, 1/16]]

derived = [203/16, 833/64, 21/64, 6]

(0)^0

Taylor ORIGINAL next derivative center power0 term6 = 1

Taylor ORIGINAL next derivative falling factor times power term6 := mul(720, 1)

Taylor ORIGINAL next derivative falling factor times power term6 = 720

Taylor ORIGINAL next derivative ORIGINAL coefficient term6 := mul(21, 720)

Taylor ORIGINAL next derivative ORIGINAL coefficient term6 = 15120

Taylor ORIGINAL next derivative ascending derivative sum term6 := sum(0, 15120)

Taylor ORIGINAL next derivative ascending derivative sum term6 = 15120

source roles = [6, 0, [4, 7, 10, 13, 16, 2, 21]]

derived = [[[6, 21, 720, 1, 15120]], 15120]

(1/2)^6

Taylor bound ORIGINAL absolute displacement power = 1/64

Taylor bound supplied M times power := mul(15119, 1/64)

Taylor bound supplied M times power = 15119/64

Taylor bound divide next factorial := div(15119/64, 720)

Taylor bound divide next factorial = 15119/46080

source roles = [undersized_M, 21/64, 15120, 15119, 1/2]

derived = [15119/46080, 否, 否, 否]

逐步反向核對

• S172 · 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。
• S171 · premise_and_result_audit — 以精確有理數重算 Taylor bound divide next factorial，不以小數近似替代。
• S170 · premise_and_result_audit — 以精確有理數重算 Taylor bound supplied M times power，不以小數近似替代。
• S169 · 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。
• S168 · 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。
• S167 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative ascending derivative sum term6，不以小數近似替代。
• S166 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative ORIGINAL coefficient term6，不以小數近似替代。
• S165 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL next derivative falling factor times power term6，不以小數近似替代。
• S164 · 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。
• S163 · 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。
• S162 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL f(x) minus Taylor value，不以小數近似替代。
• S161 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum6，不以小數近似替代。
• S160 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term6，不以小數近似替代。
• S159 · 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。
• S158 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum5，不以小數近似替代。
• S157 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term5，不以小數近似替代。
• S156 · 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。
• S155 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum4，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term4，不以小數近似替代。
• S153 · 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。
• S152 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum3，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term3，不以小數近似替代。
• S150 · 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。
• S149 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum2，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term2，不以小數近似替代。
• S147 · 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。
• S146 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum1，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term1，不以小數近似替代。
• S144 · 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。
• S143 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum0，不以小數近似替代。
• S142 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL full function at x coefficient term0，不以小數近似替代。
• S141 · 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。
• S140 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum5，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum4，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum3，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum2，不以小數近似替代。
• S136 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum1，不以小數近似替代。
• S135 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL partial sum0，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term5 divide factorial，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement5，不以小數近似替代。
• S132 · 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。
• S131 · 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。
• S130 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 ascending derivative sum term6，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 ORIGINAL coefficient term6，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 falling factor times power term6，不以小數近似替代。
• S127 · 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。
• S126 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 ascending derivative sum term5，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 ORIGINAL coefficient term5，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative5 falling factor times power term5，不以小數近似替代。
• S123 · 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。
• S122 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term4 divide factorial，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement4，不以小數近似替代。
• S120 · 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。
• S119 · 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。
• S118 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ascending derivative sum term6，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ORIGINAL coefficient term6，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 falling factor times power term6，不以小數近似替代。
• S115 · 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。
• S114 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ascending derivative sum term5，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ORIGINAL coefficient term5，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 falling factor times power term5，不以小數近似替代。
• S111 · 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。
• S110 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ascending derivative sum term4，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 ORIGINAL coefficient term4，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative4 falling factor times power term4，不以小數近似替代。
• S107 · 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。
• S106 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term3 divide factorial，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement3，不以小數近似替代。
• S104 · 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。
• S103 · 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。
• S102 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term6，不以小數近似替代。
• S101 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term6，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term6，不以小數近似替代。
• S099 · 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。
• S098 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term5，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term5，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term5，不以小數近似替代。
• S095 · 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。
• S094 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term4，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term4，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term4，不以小數近似替代。
• S091 · 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。
• S090 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term3，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term3，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term3，不以小數近似替代。
• S087 · 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。
• S086 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term2 divide factorial，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement2，不以小數近似替代。
• S084 · 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。
• S083 · 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。
• S082 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term6，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term6，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term6，不以小數近似替代。
• S079 · 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。
• S078 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term5，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term5，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term5，不以小數近似替代。
• S075 · 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。
• S074 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term4，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term4，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term4，不以小數近似替代。
• S071 · 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。
• S070 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term3，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term3，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term3，不以小數近似替代。
• S067 · 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。
• S066 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term2，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term2，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term2，不以小數近似替代。
• S063 · 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。
• S062 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term1 divide factorial，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement1，不以小數近似替代。
• S060 · 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。
• S059 · 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。
• S058 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term6，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term6，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term6，不以小數近似替代。
• S055 · 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。
• S054 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term5，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term5，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term5，不以小數近似替代。
• S051 · 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。
• S050 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term4，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term4，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term4，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term3，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term3，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term3，不以小數近似替代。
• S043 · 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。
• S042 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term2，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term2，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term2，不以小數近似替代。
• S039 · 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。
• S038 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term1，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term1，不以小數近似替代。
• S035 · 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。
• S034 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL term0 divide factorial，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative times displacement0，不以小數近似替代。
• 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 Taylor ORIGINAL derivative0 ascending derivative sum term6，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term6，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term6，不以小數近似替代。
• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term5，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term5，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term5，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term4，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term4，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term4，不以小數近似替代。
• S019 · 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。
• S018 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term3，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term3，不以小數近似替代。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term2，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term2，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 Taylor ORIGINAL derivative0 ascending derivative sum term1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term1，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term0，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 — 以精確有理數重算 Taylor ORIGINAL x minus declared center，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

(undersized_M, 21/64, 15120, 15119, 15119/46080, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)

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。