第 24234 題：實指數函數

原題與已知條件

E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[9,10],"cutoff":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-finite-partial-sum；資料={"x":[9,10],"cutoff":13}。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-real-exponential-function

逐步推導

source roles = [exp-finite-partial-sum]

derived = [[x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary]]

source roles = [0]

derived = [1]

(9/10)^0

Exp ORIGINAL x power0 = 1

Exp term0 power over factorial := div(1, 1)

Exp term0 power over factorial = 1

Exp cumulative sum through0 := sum(0, 1)

Exp cumulative sum through0 = 1

Exp term1 factorial step1 := mul(1, 1)

Exp term1 factorial step1 = 1

(9/10)^1

Exp ORIGINAL x power1 = 9/10

Exp term1 power over factorial := div(9/10, 1)

Exp term1 power over factorial = 9/10

Exp cumulative sum through1 := sum(1, 9/10)

Exp cumulative sum through1 = 19/10

Exp term2 factorial step1 := mul(1, 1)

Exp term2 factorial step1 = 1

Exp term2 factorial step2 := mul(1, 2)

Exp term2 factorial step2 = 2

(9/10)^2

Exp ORIGINAL x power2 = 81/100

Exp term2 power over factorial := div(81/100, 2)

Exp term2 power over factorial = 81/200

Exp cumulative sum through2 := sum(19/10, 81/200)

Exp cumulative sum through2 = 461/200

Exp term3 factorial step1 := mul(1, 1)

Exp term3 factorial step1 = 1

Exp term3 factorial step2 := mul(1, 2)

Exp term3 factorial step2 = 2

Exp term3 factorial step3 := mul(2, 3)

Exp term3 factorial step3 = 6

(9/10)^3

Exp ORIGINAL x power3 = 729/1000

Exp term3 power over factorial := div(729/1000, 6)

Exp term3 power over factorial = 243/2000

Exp cumulative sum through3 := sum(461/200, 243/2000)

Exp cumulative sum through3 = 4853/2000

Exp term4 factorial step1 := mul(1, 1)

Exp term4 factorial step1 = 1

Exp term4 factorial step2 := mul(1, 2)

Exp term4 factorial step2 = 2

Exp term4 factorial step3 := mul(2, 3)

Exp term4 factorial step3 = 6

Exp term4 factorial step4 := mul(6, 4)

Exp term4 factorial step4 = 24

(9/10)^4

Exp ORIGINAL x power4 = 6561/10000

Exp term4 power over factorial := div(6561/10000, 24)

Exp term4 power over factorial = 2187/80000

Exp cumulative sum through4 := sum(4853/2000, 2187/80000)

Exp cumulative sum through4 = 196307/80000

Exp term5 factorial step1 := mul(1, 1)

Exp term5 factorial step1 = 1

Exp term5 factorial step2 := mul(1, 2)

Exp term5 factorial step2 = 2

Exp term5 factorial step3 := mul(2, 3)

Exp term5 factorial step3 = 6

Exp term5 factorial step4 := mul(6, 4)

Exp term5 factorial step4 = 24

Exp term5 factorial step5 := mul(24, 5)

Exp term5 factorial step5 = 120

(9/10)^5

Exp ORIGINAL x power5 = 59049/100000

Exp term5 power over factorial := div(59049/100000, 120)

Exp term5 power over factorial = 19683/4000000

Exp cumulative sum through5 := sum(196307/80000, 19683/4000000)

Exp cumulative sum through5 = 9835033/4000000

Exp term6 factorial step1 := mul(1, 1)

Exp term6 factorial step1 = 1

Exp term6 factorial step2 := mul(1, 2)

Exp term6 factorial step2 = 2

Exp term6 factorial step3 := mul(2, 3)

Exp term6 factorial step3 = 6

Exp term6 factorial step4 := mul(6, 4)

Exp term6 factorial step4 = 24

Exp term6 factorial step5 := mul(24, 5)

Exp term6 factorial step5 = 120

Exp term6 factorial step6 := mul(120, 6)

Exp term6 factorial step6 = 720

(9/10)^6

Exp ORIGINAL x power6 = 531441/1000000

Exp term6 power over factorial := div(531441/1000000, 720)

Exp term6 power over factorial = 59049/80000000

Exp cumulative sum through6 := sum(9835033/4000000, 59049/80000000)

Exp cumulative sum through6 = 196759709/80000000

Exp term7 factorial step1 := mul(1, 1)

Exp term7 factorial step1 = 1

Exp term7 factorial step2 := mul(1, 2)

Exp term7 factorial step2 = 2

Exp term7 factorial step3 := mul(2, 3)

Exp term7 factorial step3 = 6

Exp term7 factorial step4 := mul(6, 4)

Exp term7 factorial step4 = 24

Exp term7 factorial step5 := mul(24, 5)

Exp term7 factorial step5 = 120

Exp term7 factorial step6 := mul(120, 6)

Exp term7 factorial step6 = 720

Exp term7 factorial step7 := mul(720, 7)

Exp term7 factorial step7 = 5040

(9/10)^7

Exp ORIGINAL x power7 = 4782969/10000000

Exp term7 power over factorial := div(4782969/10000000, 5040)

Exp term7 power over factorial = 531441/5600000000

Exp cumulative sum through7 := sum(196759709/80000000, 531441/5600000000)

Exp cumulative sum through7 = 13773711071/5600000000

Exp term8 factorial step1 := mul(1, 1)

Exp term8 factorial step1 = 1

Exp term8 factorial step2 := mul(1, 2)

Exp term8 factorial step2 = 2

Exp term8 factorial step3 := mul(2, 3)

Exp term8 factorial step3 = 6

Exp term8 factorial step4 := mul(6, 4)

Exp term8 factorial step4 = 24

Exp term8 factorial step5 := mul(24, 5)

Exp term8 factorial step5 = 120

Exp term8 factorial step6 := mul(120, 6)

Exp term8 factorial step6 = 720

Exp term8 factorial step7 := mul(720, 7)

Exp term8 factorial step7 = 5040

Exp term8 factorial step8 := mul(5040, 8)

Exp term8 factorial step8 = 40320

(9/10)^8

Exp ORIGINAL x power8 = 43046721/100000000

Exp term8 power over factorial := div(43046721/100000000, 40320)

Exp term8 power over factorial = 4782969/448000000000

Exp cumulative sum through8 := sum(13773711071/5600000000, 4782969/448000000000)

Exp cumulative sum through8 = 1101901668649/448000000000

Exp term9 factorial step1 := mul(1, 1)

Exp term9 factorial step1 = 1

Exp term9 factorial step2 := mul(1, 2)

Exp term9 factorial step2 = 2

Exp term9 factorial step3 := mul(2, 3)

Exp term9 factorial step3 = 6

Exp term9 factorial step4 := mul(6, 4)

Exp term9 factorial step4 = 24

Exp term9 factorial step5 := mul(24, 5)

Exp term9 factorial step5 = 120

Exp term9 factorial step6 := mul(120, 6)

Exp term9 factorial step6 = 720

Exp term9 factorial step7 := mul(720, 7)

Exp term9 factorial step7 = 5040

Exp term9 factorial step8 := mul(5040, 8)

Exp term9 factorial step8 = 40320

Exp term9 factorial step9 := mul(40320, 9)

Exp term9 factorial step9 = 362880

(9/10)^9

Exp ORIGINAL x power9 = 387420489/1000000000

Exp term9 power over factorial := div(387420489/1000000000, 362880)

Exp term9 power over factorial = 4782969/4480000000000

Exp cumulative sum through9 := sum(1101901668649/448000000000, 4782969/4480000000000)

Exp cumulative sum through9 = 11019021469459/4480000000000

Exp term10 factorial step1 := mul(1, 1)

Exp term10 factorial step1 = 1

Exp term10 factorial step2 := mul(1, 2)

Exp term10 factorial step2 = 2

Exp term10 factorial step3 := mul(2, 3)

Exp term10 factorial step3 = 6

Exp term10 factorial step4 := mul(6, 4)

Exp term10 factorial step4 = 24

Exp term10 factorial step5 := mul(24, 5)

Exp term10 factorial step5 = 120

Exp term10 factorial step6 := mul(120, 6)

Exp term10 factorial step6 = 720

Exp term10 factorial step7 := mul(720, 7)

Exp term10 factorial step7 = 5040

Exp term10 factorial step8 := mul(5040, 8)

Exp term10 factorial step8 = 40320

Exp term10 factorial step9 := mul(40320, 9)

Exp term10 factorial step9 = 362880

Exp term10 factorial step10 := mul(362880, 10)

Exp term10 factorial step10 = 3628800

(9/10)^10

Exp ORIGINAL x power10 = 3486784401/10000000000

Exp term10 power over factorial := div(3486784401/10000000000, 3628800)

Exp term10 power over factorial = 43046721/448000000000000

Exp cumulative sum through10 := sum(11019021469459/4480000000000, 43046721/448000000000000)

Exp cumulative sum through10 = 1101902189992621/448000000000000

Exp term11 factorial step1 := mul(1, 1)

Exp term11 factorial step1 = 1

Exp term11 factorial step2 := mul(1, 2)

Exp term11 factorial step2 = 2

Exp term11 factorial step3 := mul(2, 3)

Exp term11 factorial step3 = 6

Exp term11 factorial step4 := mul(6, 4)

Exp term11 factorial step4 = 24

Exp term11 factorial step5 := mul(24, 5)

Exp term11 factorial step5 = 120

Exp term11 factorial step6 := mul(120, 6)

Exp term11 factorial step6 = 720

Exp term11 factorial step7 := mul(720, 7)

Exp term11 factorial step7 = 5040

Exp term11 factorial step8 := mul(5040, 8)

Exp term11 factorial step8 = 40320

Exp term11 factorial step9 := mul(40320, 9)

Exp term11 factorial step9 = 362880

Exp term11 factorial step10 := mul(362880, 10)

Exp term11 factorial step10 = 3628800

Exp term11 factorial step11 := mul(3628800, 11)

Exp term11 factorial step11 = 39916800

(9/10)^11

Exp ORIGINAL x power11 = 31381059609/100000000000

Exp term11 power over factorial := div(31381059609/100000000000, 39916800)

Exp term11 power over factorial = 387420489/49280000000000000

Exp cumulative sum through11 := sum(1101902189992621/448000000000000, 387420489/49280000000000000)

Exp cumulative sum through11 = 121209241286608799/49280000000000000

Exp term12 factorial step1 := mul(1, 1)

Exp term12 factorial step1 = 1

Exp term12 factorial step2 := mul(1, 2)

Exp term12 factorial step2 = 2

Exp term12 factorial step3 := mul(2, 3)

Exp term12 factorial step3 = 6

Exp term12 factorial step4 := mul(6, 4)

Exp term12 factorial step4 = 24

Exp term12 factorial step5 := mul(24, 5)

Exp term12 factorial step5 = 120

Exp term12 factorial step6 := mul(120, 6)

Exp term12 factorial step6 = 720

Exp term12 factorial step7 := mul(720, 7)

Exp term12 factorial step7 = 5040

Exp term12 factorial step8 := mul(5040, 8)

Exp term12 factorial step8 = 40320

Exp term12 factorial step9 := mul(40320, 9)

Exp term12 factorial step9 = 362880

Exp term12 factorial step10 := mul(362880, 10)

Exp term12 factorial step10 = 3628800

Exp term12 factorial step11 := mul(3628800, 11)

Exp term12 factorial step11 = 39916800

Exp term12 factorial step12 := mul(39916800, 12)

Exp term12 factorial step12 = 479001600

(9/10)^12

Exp ORIGINAL x power12 = 282429536481/1000000000000

Exp term12 power over factorial := div(282429536481/1000000000000, 479001600)

Exp term12 power over factorial = 1162261467/1971200000000000000

Exp cumulative sum through12 := sum(121209241286608799/49280000000000000, 1162261467/1971200000000000000)

Exp cumulative sum through12 = 4848369652626613427/1971200000000000000

Exp term13 factorial step1 := mul(1, 1)

Exp term13 factorial step1 = 1

Exp term13 factorial step2 := mul(1, 2)

Exp term13 factorial step2 = 2

Exp term13 factorial step3 := mul(2, 3)

Exp term13 factorial step3 = 6

Exp term13 factorial step4 := mul(6, 4)

Exp term13 factorial step4 = 24

Exp term13 factorial step5 := mul(24, 5)

Exp term13 factorial step5 = 120

Exp term13 factorial step6 := mul(120, 6)

Exp term13 factorial step6 = 720

Exp term13 factorial step7 := mul(720, 7)

Exp term13 factorial step7 = 5040

Exp term13 factorial step8 := mul(5040, 8)

Exp term13 factorial step8 = 40320

Exp term13 factorial step9 := mul(40320, 9)

Exp term13 factorial step9 = 362880

Exp term13 factorial step10 := mul(362880, 10)

Exp term13 factorial step10 = 3628800

Exp term13 factorial step11 := mul(3628800, 11)

Exp term13 factorial step11 = 39916800

Exp term13 factorial step12 := mul(39916800, 12)

Exp term13 factorial step12 = 479001600

Exp term13 factorial step13 := mul(479001600, 13)

Exp term13 factorial step13 = 6227020800

(9/10)^13

Exp ORIGINAL x power13 = 2541865828329/10000000000000

Exp term13 power over factorial := div(2541865828329/10000000000000, 6227020800)

Exp term13 power over factorial = 10460353203/256256000000000000000

Exp cumulative sum through13 := sum(4848369652626613427/1971200000000000000, 10460353203/256256000000000000000)

Exp cumulative sum through13 = 630288054851920098713/256256000000000000000

source roles = [9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000]]

derived = [630288054851920098713/256256000000000000000, 否, finite sum is not a general infinite-value certificate]

逐步反向核對

• S136 · 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。
• S135 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through13，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Exp term13 power over factorial，不以小數近似替代。
• S133 · 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。
• S132 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step13，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step12，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step11，不以小數近似替代。
• S129 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step10，不以小數近似替代。
• S128 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step9，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step8，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step7，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step6，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step5，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step4，不以小數近似替代。
• S122 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step3，不以小數近似替代。
• S121 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step2，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Exp term13 factorial step1，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through12，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Exp term12 power over factorial，不以小數近似替代。
• S117 · 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。
• S116 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step12，不以小數近似替代。
• S115 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step11，不以小數近似替代。
• S114 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step10，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step9，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step8，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step7，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step6，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step5，不以小數近似替代。
• S108 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step4，不以小數近似替代。
• S107 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step3，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step2，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Exp term12 factorial step1，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through11，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Exp term11 power over factorial，不以小數近似替代。
• S102 · 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。
• S101 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step11，不以小數近似替代。
• S100 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step10，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step9，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step8，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step7，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step6，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step5，不以小數近似替代。
• S094 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step4，不以小數近似替代。
• S093 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step3，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step2，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Exp term11 factorial step1，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through10，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Exp term10 power over factorial，不以小數近似替代。
• S088 · 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。
• S087 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step10，不以小數近似替代。
• S086 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step9，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step8，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step7，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step6，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step5，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step4，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step3，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step2，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Exp term10 factorial step1，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through9，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Exp term9 power over factorial，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 Exp term9 factorial step9，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step8，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step7，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step6，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step5，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step4，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step3，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step2，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Exp term9 factorial step1，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through8，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Exp term8 power over factorial，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 Exp term8 factorial step8，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step7，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step6，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step5，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step4，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step3，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step2，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Exp term8 factorial step1，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through7，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Exp term7 power over factorial，不以小數近似替代。
• S052 · 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。
• S051 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step7，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step6，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step5，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step4，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step3，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step2，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Exp term7 factorial step1，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through6，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Exp term6 power over factorial，不以小數近似替代。
• S042 · 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。
• S041 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step6，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step5，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step4，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step3，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step2，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Exp term6 factorial step1，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through5，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Exp term5 power over factorial，不以小數近似替代。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step5，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step4，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step2，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Exp term5 factorial step1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through4，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Exp term4 power over factorial，不以小數近似替代。
• S025 · 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。
• S024 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step4，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step3，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step2，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Exp term4 factorial step1，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through3，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Exp term3 power over factorial，不以小數近似替代。
• S018 · 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。
• S017 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Exp term3 factorial step1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Exp term2 power over factorial，不以小數近似替代。
• S012 · 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。
• S011 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step2，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Exp term2 factorial step1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Exp term1 power over factorial，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 Exp term1 factorial step1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Exp cumulative sum through0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Exp term0 power 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

(9/10, 13, [1, 9/10, 81/200, 243/2000, 2187/80000, 19683/4000000, 59049/80000000, 531441/5600000000, 4782969/448000000000, 4782969/4480000000000, 43046721/448000000000000, 387420489/49280000000000000, 1162261467/1971200000000000000, 10460353203/256256000000000000000], [1, 19/10, 461/200, 4853/2000, 196307/80000, 9835033/4000000, 196759709/80000000, 13773711071/5600000000, 1101901668649/448000000000, 11019021469459/4480000000000, 1101902189992621/448000000000000, 121209241286608799/49280000000000000, 4848369652626613427/1971200000000000000, 630288054851920098713/256256000000000000000], 630288054851920098713/256256000000000000000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)

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。