實指數函數基礎 · 脈衝 R02
本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。
本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。
batch-20261015-advance-20261113-r02-real-exponential-function-foundations · 2026-10-15 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
欄序與來源綁定資料
1. 實指數函數
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":[1,2],"cutoff":5}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(1/2, 5, [1, 1/2, 1/8, 1/48, 1/384, 1/3840], [1, 3/2, 13/8, 79/48, 211/128, 6331/3840], 6331/3840, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
2. 實指數函數
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":[2,3],"cutoff":6}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(2/3, 6, [1, 2/3, 2/9, 4/81, 2/243, 4/3645, 4/32805], [1, 5/3, 17/9, 157/81, 473/243, 7099/3645, 12779/6561], 12779/6561, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
3. 實指數函數
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":[3,4],"cutoff":7}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(3/4, 7, [1, 3/4, 9/32, 9/128, 27/2048, 81/40960, 81/327680, 243/9175040], [1, 7/4, 65/32, 269/128, 4331/2048, 86701/40960, 693689/327680, 3884707/1835008], 3884707/1835008, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
4. 實指數函數
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":[4,5],"cutoff":8}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(4/5, 8, [1, 4/5, 8/25, 32/375, 32/1875, 128/46875, 256/703125, 1024/24609375, 512/123046875], [1, 9/5, 53/25, 827/375, 1389/625, 104303/46875, 1564801/703125, 6085451/2734375, 273845807/123046875], 273845807/123046875, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
5. 實指數函數
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":[5,6],"cutoff":9}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(5/6, 9, [1, 5/6, 25/72, 125/1296, 625/31104, 625/186624, 3125/6718464, 15625/282175488, 78125/13544423424, 390625/731398864896], [1, 11/6, 157/72, 2951/1296, 71449/31104, 429319/186624, 15458609/6718464, 649277203/282175488, 31165383869/13544423424, 1682931119551/731398864896], 1682931119551/731398864896, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
6. 實指數函數
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":[6,7],"cutoff":10}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(6/7, 10, [1, 6/7, 18/49, 36/343, 54/2401, 324/84035, 324/588245, 1944/28824005, 1458/201768035, 972/1412376245, 2916/49433168575], [1, 13/7, 109/49, 799/343, 5647/2401, 197969/84035, 1386107/588245, 67921187/28824005, 475449767/201768035, 3328149341/1412376245, 116485229851/49433168575], 116485229851/49433168575, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
7. 實指數函數
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":[7,8],"cutoff":11}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(7/8, 11, [1, 7/8, 49/128, 343/3072, 2401/98304, 16807/3932160, 117649/188743680, 117649/1509949440, 823543/96636764160, 5764801/6957847019520, 40353607/556627761561600, 282475249/48983243017420800], [1, 15/8, 289/128, 7279/3072, 78443/32768, 9429967/3932160, 90551213/37748736, 1207388723/503316480, 231819458359/96636764160, 16691006766649/6957847019520, 445093527228509/185542587187200, 4700187658832065/1959329720696832], 4700187658832065/1959329720696832, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
8. 實指數函數
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":[8,9],"cutoff":12}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(8/9, 12, [1, 8/9, 32/81, 256/2187, 512/19683, 4096/885735, 16384/23914845, 131072/1506635235, 131072/13559717115, 1048576/1098337086315, 4194304/49425168884175, 33554432/4893091719533325, 67108864/132113476427399775], [1, 17/9, 185/81, 5251/2187, 47771/19683, 2153791/885735, 58168741/23914845, 732952351/301327047, 32982986867/13559717115, 2671622984803/1098337086315, 120223038510439/49425168884175, 11902080846087893/4893091719533325, 12854247316459279/5284539057095991], 12854247316459279/5284539057095991, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
9. 實指數函數
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 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(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)
Wave6第七層核對=(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)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
10. 實指數函數
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":[10,11],"cutoff":14}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(10/11, 14, [1, 10/11, 50/121, 500/3993, 1250/43923, 2500/483153, 12500/15944049, 125000/1227691773, 156250/13504609503, 1562500/1336956340797, 1562500/14706519748767, 15625000/1779488889600807, 39062500/58723133356826631, 390625000/8397408070026208233, 1953125000/646600421392018033941], [1, 21/11, 281/121, 9773/3993, 36251/14641, 1198783/483153, 39572339/15944049, 1015731701/409230591, 4788471769/1929229929, 3318412498417/1336956340797, 12167513015029/4902173249589, 4416807240080527/1779488889600807, 145754638961719891/58723133356826631, 6947637790638856471/2799136023342069411, 1604904329639528969801/646600421392018033941], 1604904329639528969801/646600421392018033941, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
11. 實指數函數
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":[11,12],"cutoff":15}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(11/12, 15, [1, 11/12, 121/288, 1331/10368, 14641/497664, 161051/29859840, 1771561/2149908480, 19487171/180592312320, 214358881/17336861982720, 2357947691/1872381094133760, 25937424601/224685731296051200, 25937424601/2696228775552614400, 285311670611/388256943679576473600, 3138428376721/60568083214013929881600, 34522712143931/10175437979954340220108800, 379749833583241/1831578836391781239619584000], [1, 23/12, 673/288, 25559/10368, 1241473/497664, 74649431/29859840, 5376530593/2149908480, 451648056983/180592312320, 43358427829249/17336861982720, 4682712563506583/1872381094133760, 561925533558214561/224685731296051200, 6743106428635999333/2696228775552614400, 971007326008895574563/388256943679576473600, 21639591837218019715507/8652583316287704268800, 25448160000602913897580163/10175437979954340220108800, 4580668800108904251398012581/1831578836391781239619584000], 4580668800108904251398012581/1831578836391781239619584000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
12. 實指數函數
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":[12,13],"cutoff":16}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(12/13, 16, [1, 12/13, 72/169, 288/2197, 864/28561, 10368/1856465, 20736/24134045, 248832/2196198095, 373248/28550575235, 497664/371157478055, 2985984/24125236073575, 35831808/3449908758521225, 35831808/44848813860775925, 429981696/7579449542471131325, 2579890176/689729908364872950575, 10319560704/44832444043716741787375, 7739670528/582821772568317643235875], [1, 25/13, 397/169, 5449/2197, 71701/28561, 4670933/1856465, 12148573/4826809, 5527849547/2196198095, 71862417359/28550575235, 934211923331/371157478055, 60723778002499/24125236073575, 248100008291119/98568821672035, 112885503808290953/44848813860775925, 19077650144031152753/7579449542471131325, 1736066163109414790699/689729908364872950575, 112844300602122280956139/44832444043716741787375, 293395181565519478420067/116564354513663528647175], 293395181565519478420067/116564354513663528647175, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
13. 實指數函數
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":[13,14],"cutoff":17}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(13/14, 17, [1, 13/14, 169/392, 2197/16464, 28561/921984, 371293/64538880, 4826809/5421265920, 62748517/531284060160, 815730721/59503814737920, 10604499373/7497480656977920, 137858491849/1049647291976908800, 1792160394037/161645682964443955200, 23298085122481/27156474738026584473600, 23298085122481/380190646332372182630400, 302875106592253/74517366681144947795558400, 3937376385699289/15648647003040439037067264000, 51185893014090757/3505296928681058344303067136000, 665416609183179841/834260669026091885944129978368000], [1, 27/14, 925/392, 41047/16464, 775731/307328, 163274803/64538880, 13719910261/5421265920, 29880310091/11806312448, 150597578589361/59503814737920, 18975305506758859/7497480656977920, 98390478103878967/38875825628774400, 409107609748089138823/161645682964443955200, 13746015692195412088949/5431294947605316894720, 106913455386330770149879/42243405148041353625600, 188595335301790353650978809/74517366681144947795558400, 3600456401216355603917386289/1422604273003676276097024000, 985724952510794599327272656317/389477436520117593811451904000, 422284569655624539435125442602171/166852133805218377188825995673600], 422284569655624539435125442602171/166852133805218377188825995673600, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
14. 實指數函數
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":[14,15],"cutoff":18}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(14/15, 18, [1, 14/15, 98/225, 1372/10125, 4802/151875, 67228/11390625, 470596/512578125, 941192/7688671875, 1647086/115330078125, 23059204/15569560546875, 161414428/1167717041015625, 2259801992/192673311767578125, 7909306972/8670299029541015625, 110730297608/1690708310760498046875, 110730297608/25360624661407470703125, 1550224166512/5706140548816680908203125, 1356446145698/85592108232250213623046875, 18990246039772/21825987599223804473876953125, 132931722278404/2946508325895213603973388671875], [1, 29/15, 533/225, 25357/10125, 385157/151875, 28954003/11390625, 1303400731/512578125, 19551952157/7688671875, 293280929441/115330078125, 39592948533739/15569560546875, 2969471301444853/1167717041015625, 489962766998202737/192673311767578125, 22048324522828430137/8670299029541015625, 4299423282062274174323/1690708310760498046875, 64491349231044842912453/25360624661407470703125, 1116196428998972298420649/438933888370513916015625, 217658303654800954638172253/85592108232250213623046875, 55502867431974262422979964287/21825987599223804473876953125, 576375931024348120002616727473/226654486607324123382568359375], 576375931024348120002616727473/226654486607324123382568359375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
15. 實指數函數
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":[15,16],"cutoff":19}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(15/16, 19, [1, 15/16, 225/512, 1125/8192, 16875/524288, 50625/8388608, 253125/268435456, 3796875/30064771072, 56953125/3848290697216, 94921875/61572651155456, 284765625/1970324836974592, 4271484375/346777171307528192, 21357421875/22193738963681804288, 320361328125/4616297704445815291904, 4805419921875/1034050685795862625386496, 4805419921875/16544810972733802006183936, 72081298828125/4235471609019853313583087616, 1081219482421875/1152048277653400101294599831552, 1802032470703125/36865544884908803241427194609664, 27030487060546875/11207125645012276185393867161337856], [1, 31/16, 1217/512, 20597/8192, 1335083/524288, 21411953/8388608, 685435621/268435456, 76772586427/30064771072, 9826948015781/3848290697216, 157231263174371/61572651155456, 5031400706345497/1970324836974592, 885526528588291847/346777171307528192, 56673697851008100083/22193738963681804288, 11788129153330046145389/4616297704445815291904, 2640540930350735756489011/1034050685795862625386496, 42248654885616577523746051/16544810972733802006183936, 10815655650717915927377817181/4235471609019853313583087616, 2941858336995274213466248695107/1152048277653400101294599831552, 94139466783848776632952428946549/36865544884908803241427194609664, 2201415223253079086419078881561367/862086588077867398876451320102912], 2201415223253079086419078881561367/862086588077867398876451320102912, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
16. 實指數函數
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":[16,17],"cutoff":20}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(16/17, 20, [1, 16/17, 128/289, 2048/14739, 8192/250563, 131072/21297855, 1048576/1086190605, 16777216/129256681995, 33554432/2197363593915, 536870912/336196629868995, 4294967296/28576713538864575, 68719476736/5343845431767675525, 274877906944/272536117020151451775, 4398046511104/60230481861453470842275, 35184372088832/7167427341512963030230725, 562949953421312/1827693972085805572708834875, 562949953421312/31070797525458694736050192875, 9007199254740992/8979460484857562778718505740875, 72057594037927936/1373857454183207105143931378353875, 1152921504606846976/443755957701175894961489835208301625, 4611686018427387904/37719256404599951071726635992705638125], [1, 33/17, 689/289, 37187/14739, 213457/83521, 54562607/21297855, 2783741533/1086190605, 110427339881/43085560665, 5631827888363/2197363593915, 861670203790451/336196629868995, 24413990539051877/9525571179621525, 13696248761127579733/5343845431767675525, 63500789735671315757/24776010638195586525, 51456806617271671705457/20076827287151156947425, 18370079962401171170936981/7167427341512963030230725, 360336183877912430657103959/140591844006600428669910375, 2413160504152097276520405947/941539318953293779880308875, 23014311728098560733374366257531/8979460484857562778718505740875, 3521189694399079864263872075330179/1373857454183207105143931378353875, 379114757096967599103384061646164931/147918652567058631653829945069433875, 8788569369066067070543147430745404119/3429023309509086461066057817518694375], 8788569369066067070543147430745404119/3429023309509086461066057817518694375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
17. 實指數函數
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":[17,18],"cutoff":21}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(17/18, 21, [1, 17/18, 289/648, 4913/34992, 83521/2519424, 1419857/226748160, 24137569/24488801280, 410338673/3085588961280, 6975757441/444324810424320, 118587876497/71980619288739840, 2015993900449/12956511471973171200, 34271896307633/2565389271450687897600, 582622237229761/554124082633348585881600, 9904578032905937/129665035336203569096294400, 168377826559400929/32675588904723299412266188800, 2862423051509815793/8822409004275290841311870976000, 48661191875666868481/2540853793231283762297818841088000, 48661191875666868481/45735368278163107721360739139584000, 827240261886336764177/14818259322124846901720879481225216000, 14063084452067724991009/5067844688166697640388540782579023872000, 239072435685151324847153/1824424087740011150539874681728448593920000, 4064231406647572522401601/689632305165724214904072629693353568501760000], [1, 35/18, 1549/648, 88559/34992, 6459769/2519424, 582799067/226748160, 12593287361/4897760256, 7934181376103/3085588961280, 163218441988039/63474972917760, 185089831802312723/71980619288739840, 33316171740410190589/12956511471973171200, 1319320407774622808851/513077854290137579520, 1424866040979214870788841/554124082633348585881600, 333418653599040857797494731/129665035336203569096294400, 84021500707126673991528073141/32675588904723299412266188800, 462975616141368661240083460487/180049163352556955945140224000, 261340475799481728344477338450441/101634151729251350491912753643520, 117603214109766826416206677969566931/45735368278163107721360739139584000, 38103441371564452586091225548476449821/14818259322124846901720879481225216000, 13031376949075042798506283589646670829791/5067844688166697640388540782579023872000, 426481427424274127972848593450722983961083/165856735249091922776352243793495326720000, 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000], 354661955046026364823033736594950747966516943/137926461033144842980814525938670713700352000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
18. 實指數函數
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":[18,19],"cutoff":22}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(18/19, 22, [1, 18/19, 162/361, 972/6859, 4374/130321, 78732/12380495, 236196/235229405, 4251528/31285510865, 9565938/594424706435, 19131876/11294069422265, 172186884/1072936595115175, 3099363912/224243748379071575, 4649045868/4260631219202359925, 83682825624/1052375911142982901475, 753145430616/139965996182016725896175, 4518872583696/13296769637291588960136625, 5083731656658/252638623108540190242595875, 91507169819844/81602275264058481448358467625, 91507169819844/1550443230017111147518810884875, 1647129056757192/559710006036177124254290729439875, 7412080755407364/53172450573436826804157619296788125, 44472484532444184/7071935926267097964952963366472820625, 400252360791997656/1478034608589823474675169343592819510625], [1, 37/19, 865/361, 17407/6859, 335107/130321, 31913897/12380495, 606600239/235229405, 16136416663/6257102173, 1532969148923/594424706435, 29126432961413/11294069422265, 2767011303521119/1072936595115175, 578305365535277783/224243748379071575, 2197560389963864749/852126243840471985, 387712440241293684377/150339415877568985925, 360960281865397565585603/139965996182016725896175, 34291226777217287603215981/13296769637291588960136625, 50117946828241042168673869/19433740239118476172507375, 8417810349271369102937255831/3264091010562339257934338705, 3998459915903900415402366339569/1550443230017111147518810884875, 1443444029641308051607383305341601/559710006036177124254290729439875, 19589597545132037844301927823265637/7596064367633832400593945613826875, 18237915314517927233089567287992752231/7071935926267097964952963366472820625, 762344860146849358343223963110255442787/295606921717964694935033868718563902125], 762344860146849358343223963110255442787/295606921717964694935033868718563902125, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
19. 實指數函數
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":[19,20],"cutoff":23}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(19/20, 23, [1, 19/20, 361/800, 6859/48000, 130321/3840000, 2476099/384000000, 47045881/46080000000, 893871739/6451200000000, 16983563041/1032192000000000, 322687697779/185794560000000000, 6131066257801/37158912000000000000, 116490258898219/8174960640000000000000, 2213314919066161/1961990553600000000000000, 42052983462257059/510117543936000000000000000, 799006685782884121/142832912302080000000000000000, 15181127029874798299/42849873690624000000000000000000, 288441413567621167681/13711959580999680000000000000000000, 5480386857784802185939/4662066257539891200000000000000000000, 104127350297911241532841/1678343852714360832000000000000000000000, 104127350297911241532841/33566877054287216640000000000000000000000, 1978419655660313589123979/13426750821714886656000000000000000000000000, 37589973457545958193355601/5639235345120252395520000000000000000000000000, 714209495693373205673756419/2481263551852911054028800000000000000000000000000, 13569980418174090907801371961/1141381233852339084853248000000000000000000000000000], [1, 39/20, 1921/800, 122119/48000, 3299947/1280000, 992460199/384000000, 119142269761/46080000000, 5560270546093/2150400000000, 2668946845687681/1032192000000000, 480410754911480359/185794560000000000, 32027385704454109867/12386304000000000000, 3019724954489995915777/1167851520000000000000, 5073137925756508057571521/1961990553600000000000000, 439671953579581692810284173/170039181312000000000000000, 28409572385203663742032429957/10987147100160000000000000000, 110797332302309469720956351630599/42849873690624000000000000000000, 11818382112246439584039866714319787/4570653193666560000000000000000000, 12054749754491373856107521833408368679/4662066257539891200000000000000000000, 619958558802413527475151165419750608183/239763407530622976000000000000000000000, 1701847024163488116679382616797575032911/658174059887984640000000000000000000000, 34717679292935157582237825038330844260508379/13426750821714886656000000000000000000000000, 1121648100233289706505959729965884657508221137/433787334240019415040000000000000000000000000, 2138609044444805707071601288300184538050899553353/827087850617637018009600000000000000000000000000, 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000], 2951280481333831875758823347834672836601149184999101/1141381233852339084853248000000000000000000000000000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
20. 實指數函數
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":[20,21],"cutoff":24}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(20/21, 24, [1, 20/21, 200/441, 4000/27783, 20000/583443, 80000/12252303, 800000/771895089, 16000000/113468578083, 40000000/2382840139743, 800000000/450356786411427, 1600000000/9457492514639967, 32000000000/2184680770881832377, 160000000000/137634888565555439751, 3200000000000/37574324578396635052023, 32000000000000/5523425713024305352647381, 128000000000000/347975819920531237216785003, 160000000000000/7307492218331155981552485063, 3200000000000000/2608774721944222685414237167491, 32000000000000000/493058422447458087543290824655799, 640000000000000000/196730310556535776929773039037663801, 640000000000000000/4131336521687251315525233819790939821, 12800000000000000000/1821919406064077830146628114527804461061, 128000000000000000000/420863382800801978763871094455922830505091, 2560000000000000000000/203277013892787355742949738622210727133958953, 6400000000000000000000/12806451875245603411805833533199275809439414039], [1, 41/21, 1061/441, 70843/27783, 1507703/583443, 31741763/12252303, 2000531069/771895089, 294094067143/113468578083, 6176015410003/2382840139743, 1167267712490567/450356786411427, 24512623562301907/9457492514639967, 5662416074891740517/2184680770881832377, 356732212878179652571/137634888565555439751, 97387894118943045151883/37574324578396635052023, 14316020435516627637326801/5523425713024305352647381, 901909287437675541151588463/347975819920531237216785003, 18940095036191346364183357723/7307492218331155981552485063, 6761613927920313852013458707111/2608774721944222685414237167491, 1277945032376939350030543695643979/493058422447458087543290824655799, 509900067918398801302186934561947621/196730310556535776929773039037663801, 10707901426286374827985925625800900041/4131336521687251315525233819790939821, 4722184528992291299154593200978196918081/1821919406064077830146628114527804461061, 1090824626197219290104839029425963488076711/420863382800801978763871094455922830505091, 526868294453256917120639811212740364741051413/203277013892787355742949738622210727133958953, 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039], 33192702550555185778600314506402642978686239019/12806451875245603411805833533199275809439414039, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
21. 實指數函數
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":[21,22],"cutoff":25}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(21/22, 25, [1, 21/22, 441/968, 3087/21296, 64827/1874048, 1361367/206145280, 9529569/9070392320, 28588707/199548631040, 600362847/35120559063040, 1400846643/772652299386880, 29417779503/169983505865113600, 617773369563/41136008419357491200, 4324413586941/3619968740903459225600, 90812685325761/1035311059898389338521600, 272438055977283/45553686635529130894950400, 1907066391840981/5010905529908204398444544000, 40048394228660601/1763838746527687948252479488000, 841016278801872621/659675691201355292646427328512000, 1962371317204369449/29025730412859632876442802454528000, 41209797661291758429/12132755312575326542353091425992704000, 865405750887126927009/5338412337533143678635360227436789760000, 865405750887126927009/117445071425729160929977925003609374720000, 18173520768629665467189/56843414570052913890109315701746937364480000, 381643936141222974810969/28762767772446774428395313745083950306426880000, 2671507552988560823676783/5062247127950632299397575219134775253931130880000, 56101658612759777297212443/2784235920372847764668666370524126389662121984000000], [1, 43/22, 2333/968, 54413/21296, 4853171/1874048, 535210177/206145280, 23558777357/9070392320, 518321690561/199548631040, 91225217901583/35120559063040, 2006956194681469/772652299386880, 441530392247702683/169983505865113600, 106850355541717418849/41136008419357491200, 9402831291995546445653/3619968740903459225600, 2689209749601538968782519/1035311059898389338521600, 118325228982740152682408119/45553686635529130894950400, 13015775188103323861456734071/5010905529908204398444544000, 4581552866212410047626999053593/1763838746527687948252479488000, 1713500771963442198828776447916403/659675691201355292646427328512000, 75394033966391458710837480912691181/29025730412859632876442802454528000, 31514706197951629782339864682796672087/12132755312575326542353091425992704000, 13866470727098717105094946211317662645289/5338412337533143678635360227436789760000, 305062355996171776312954222399875705123367/117445071425729160929977925003609374720000, 147650180302147139735488017162308470945176817/56843414570052913890109315701746937364480000, 74710991232886452706157318328064227521234280371/28762767772446774428395313745083950306426880000, 13149134456988015676283690697246857032298057022079/5062247127950632299397575219134775253931130880000, 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000], 7232023951343408621956029939587429980523708659355893/2784235920372847764668666370524126389662121984000000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
22. 實指數函數
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":[22,23],"cutoff":26}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
23. 實指數函數
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":[23,24],"cutoff":27}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(23/24, 27, [1, 23/24, 529/1152, 12167/82944, 279841/7962624, 6436343/955514880, 148035889/137594142720, 3404825447/23115815976960, 78310985281/4438236667576320, 1801152661463/958659120196485120, 41426511213649/230078188847156428800, 952809757913927/60740641855649297203200, 21914624432020321/17493304854426997594521600, 504036361936467383/5457911114581223249490739200, 11592836324538749809/1833858134499291011828888371200, 266635235464391245607/660188928419744764258399813632000, 6132610415680998648961/253512548513181989475225528434688000, 141050039560662968926103/103433119793378251705892015601352704000, 3244150909895248285300369/44683107750739404736945350739784368128000, 74615470927590710561908487/20375497134337168560047079937341671866368000, 1716155831334586342923895201/9780238624481840908822598369924002495856640000, 39471584120695485887249589623/4929240266738847818046589578441697257911746560000, 907846434775996175406740561329/2602638860838111647928599297417216152177402183680000, 907846434775996175406740561329/62463332660114679550286383138013187652257652408320000, 20880467999847912034355032910567/35978879612226055420964956687495596087700407787192320000, 480250763996501976790165756943041/21587327767335633252578974012497357652620244672315392000000, 11045767571919545466173812409689943/13470492526817435149609279783798351175235032675524804608000000, 254052654154149545721997685422868689/8728879157377697976946813299901331561552301173740073385984000000], [1, 47/24, 2785/1152, 212687/82944, 20697793/7962624, 2490171503/955514880, 358732732321/137594142720, 12054100771075/4623163195392, 11572015051217281/4438236667576320, 2499557052215594159/958659120196485120, 599893733958253811809/230078188847156428800, 158371946717788764231503/60740641855649297203200, 9122224135327557706138637/3498660970885399518904320, 14230669651615026383512741103/5457911114581223249490739200, 4781505002954241701184819760417/1833858134499291011828888371200, 1721341801063793647661999504995727/660188928419744764258399813632000, 660995251608502893312623490917008129/253512548513181989475225528434688000, 53937212531253864304317988991421648547/20686623958675650341178403120270540800, 10591307187955304558316160556065367832899/4062100704612673157904122794525851648000, 53125996854783807739129332276814595611729871/20375497134337168560047079937341671866368000, 1500028146488013395088131489659152484503190193/575308154381284759342505786466117793873920000, 12852241159109298769154582187520313973110583163247/4929240266738847818046589578441697257911746560000, 1357196666401941950022905448289100354795558930151149/520527772167622329585719859483443230435480436736000, 162863599968233034002749561641126818571642478358699209/62463332660114679550286383138013187652257652408320000, 93809433581702227585583768385757047345178101889643654951/35978879612226055420964956687495596087700407787192320000, 56285660149021336551350261511704992403608837923951949913641/21587327767335633252578974012497357652620244672315392000000, 35122251932989314008042563194349682831771460330719829155801927/13470492526817435149609279783798351175235032675524804608000000, 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000], 4551843850515415095442316190038529425828411168005689395676503477/1745775831475539595389362659980266312310460234748014677196800000, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
24. 實指數函數
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":[24,25],"cutoff":28}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(24/25, 28, [1, 24/25, 288/625, 2304/15625, 13824/390625, 331776/48828125, 1327104/1220703125, 31850496/213623046875, 95551488/5340576171875, 254803968/133514404296875, 3057647616/16689300537109375, 73383542784/4589557647705078125, 146767085568/114738941192626953125, 3522410053632/37290155887603759765625, 42268920643584/6525777280330657958984375, 338151365148672/815722160041332244873046875, 507227047723008/20393054001033306121826171875, 12173449145352192/8667047950439155101776123046875, 16231265527136256/216676198760978877544403076171875, 389550372651270144/102921194411464966833591461181640625, 2337302235907620864/12865149301433120854198932647705078125, 18698417887260966912/2251401127750796149484813213348388671875, 224381014647131602944/619135310131468941108323633670806884765625, 5385144351531158470656/356002803325594641137286089360713958740234375, 5385144351531158470656/8900070083139866028432152234017848968505859375, 129243464436747803295744/5562543801962416267770095146261155605316162109375, 1550921573240973639548928/1807826735637785287025280922534875571727752685546875, 1378596953991976568487936/45195668390944632175632023063371889293193817138671875, 8271581723951859410927616/7909241968415310630735604036090080626308917999267578125], [1, 49/25, 1513/625, 40129/15625, 1017049/390625, 127462901/48828125, 3187899629/1220703125, 557914285571/213623046875, 1992564670109/762939453125, 348699072073043/133514404296875, 43587387066777991/16689300537109375, 11986531516747490309/4589557647705078125, 299663288065454343293/114738941192626953125, 97390568624795071623857/37290155887603759765625, 17043349509381406454818559/6525777280330657958984375, 27667775177571609846980111/10593794286251068115234375, 53260467216825856182484436683/20393054001033306121826171875, 22635698567151001051005030942467/8667047950439155101776123046875, 565892464178775042506391300697931/216676198760978877544403076171875, 15811701204995185034122720028399257/6054187906556762754917144775390625, 33599865060614768199848082296256041989/12865149301433120854198932647705078125, 452305875815968033460931755364005254999/173184702134676626883447170257568359375, 230999072291726531374722200920137988317767/88447901447352705872617661952972412109375, 929771265974199288783262243847906934137482831/356002803325594641137286089360713958740234375, 23244281649354982219581561481342024884595541431/8900070083139866028432152234017848968505859375, 44977325172900507390831195216972848265077451053/17221497838892929621579241938889026641845703125, 39020617438225047631012435697955754709648587304443/14940716823452770967151082004420459270477294921875, 118037367750630769083812617987694754950678953164428011/45195668390944632175632023063371889293193817138671875, 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125], 20656539356360384589667208147854853698092768663185829541/7909241968415310630735604036090080626308917999267578125, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
25. 實指數函數
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":[25,26],"cutoff":29}。請依 [x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary] 欄序回答。
固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。
查看答案與解釋
(25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, 否, finite_partial_sum_is_not_the_infinite_exponential_value)
Wave6第七層核對=(25/26, 29, [1, 25/26, 625/1352, 15625/105456, 390625/10967424, 1953125/285153024, 48828125/44483871744, 1220703125/8096064657408, 30517578125/1683981448740864, 762939453125/394051659005362176, 3814697265625/20490686268278833152, 95367431640625/5860336272727746281472, 2384185791015625/1828424917091056839819264, 59604644775390625/618007621976777211858911232, 1490116119384765625/224954774399546905116643688448, 7450580596923828125/17546472403164658599098207698944, 186264514923095703125/7299332519716497977224854402760704, 4656612873077392578125/3226304973714692105933385646020231168, 116415321826934814453125/1509910727698475905576824482337468186624, 2910383045673370361328125/745895899483047097354951294274709284192256, 14551915228366851806640625/77573173546236898124914934604569765555994624, 363797880709171295166015625/42354952756245346376203554294095091993573064704, 9094947017729282379150390625/24227032976572338127188433056222392620323793010688, 227373675443232059478759765625/14487765719990258200058682967620990786953628220391424, 5684341886080801486968994140625/9040365809273921116836618171795498251059064009524248576, 5684341886080801486968994140625/235049511041121949037752072466682954527535664247630462976, 142108547152020037174224853515625/158893469463798437549520400987477677260614109031398192971776, 3552713678800500929355621337890625/111543215563586503159763321493209329436951104540041531466186752, 88817841970012523233890533447265625/81203460930290974300307698047056391830100404105150234907383955456, 2220446049250313080847263336181640625/61227409541439394622432004327480519439895704695283277120167502413824], [1, 51/26, 3277/1352, 271231/105456, 9532883/3655808, 745517999/285153024, 116349635969/44483871744, 7058951483161/2698688219136, 4404816243070589/1683981448740864, 1030727763817970951/394051659005362176, 17865949177743918359/6830228756092944384, 15328984489871713592647/5860336272727746281472, 4782643163224160431921489/1828424917091056839819264, 76977780439493850988802567/29428934379846533898043392, 588418153680981113077791587773/224954774399546905116643688448, 45896615987123977400664667674419/17546472403164658599098207698944, 6364330750214586954397141616087143/2433110839905499325741618134253568, 8439102574784546958143482860324129743/3226304973714692105933385646020231168, 232323529705833417225086576798029833697/88818278099910347386872028372792246272, 650351000823196346922220039207741634905177/248631966494349032451650431424903094730752, 28987073179548180036326366780168891699579407/11081881935176699732130704943509966507999232, 110788593692233144099203171714514675370958509179/42354952756245346376203554294095091993573064704, 21123691863985786141584436389240041198189139213671/8075677658857446042396144352074130873441264336896, 37895903203990500338002706255972077141610794509091399/14487765719990258200058682967620990786953628220391424, 23647043599290072210913694388068462217166622742667173601/9040365809273921116836618171795498251059064009524248576, 10786370764588453990943088767967050942581292601374397443/4123675632300385070837755657310227272412906390309306368, 415620438301122309179019096549414954071562403690383135789301/158893469463798437549520400987477677260614109031398192971776, 5954398932395670633544314403698816559939536904489889442081223/2276392154358908227750271867208353661978593970204929213595648, 70801772905472787613264261136277667389377724326732408903789550827/27067820310096991433435899349018797276700134701716744969127985152, 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824], 160153610312179445581203758690262304080821662740149556203708145611299/61227409541439394622432004327480519439895704695283277120167502413824, 否, finite_partial_sum_is_not_the_infinite_exponential_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-finite-partial-sum · candidate
26. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":2,"start_index":2}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(2, 2, epsilon=1/2, |a_(n+1)/a_n|=1/(n+1), 1/3, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
27. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":3,"start_index":4}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(3, 4, epsilon=1/3, |a_(n+1)/a_n|=1/(n+1), 1/5, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
28. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":4,"start_index":6}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(4, 6, epsilon=1/4, |a_(n+1)/a_n|=1/(n+1), 1/7, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
29. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":5,"start_index":5}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(5, 5, epsilon=1/5, |a_(n+1)/a_n|=1/(n+1), 1/6, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
30. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":6,"start_index":7}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(6, 7, epsilon=1/6, |a_(n+1)/a_n|=1/(n+1), 1/8, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
31. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":7,"start_index":9}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(7, 9, epsilon=1/7, |a_(n+1)/a_n|=1/(n+1), 1/10, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
32. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":8,"start_index":8}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(8, 8, epsilon=1/8, |a_(n+1)/a_n|=1/(n+1), 1/9, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
33. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":9,"start_index":10}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(9, 10, epsilon=1/9, |a_(n+1)/a_n|=1/(n+1), 1/11, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
34. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":10,"start_index":12}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(10, 12, epsilon=1/10, |a_(n+1)/a_n|=1/(n+1), 1/13, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
35. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":11,"start_index":11}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(11, 11, epsilon=1/11, |a_(n+1)/a_n|=1/(n+1), 1/12, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
36. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":12,"start_index":13}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(12, 13, epsilon=1/12, |a_(n+1)/a_n|=1/(n+1), 1/14, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
37. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":13,"start_index":15}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(13, 15, epsilon=1/13, |a_(n+1)/a_n|=1/(n+1), 1/16, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
38. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":14,"start_index":14}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(14, 14, epsilon=1/14, |a_(n+1)/a_n|=1/(n+1), 1/15, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
39. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":15,"start_index":16}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(15, 16, epsilon=1/15, |a_(n+1)/a_n|=1/(n+1), 1/17, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
40. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":16,"start_index":18}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(16, 18, epsilon=1/16, |a_(n+1)/a_n|=1/(n+1), 1/19, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
41. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":17,"start_index":17}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(17, 17, epsilon=1/17, |a_(n+1)/a_n|=1/(n+1), 1/18, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
42. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":18,"start_index":19}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(18, 19, epsilon=1/18, |a_(n+1)/a_n|=1/(n+1), 1/20, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
43. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":19,"start_index":21}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(19, 21, epsilon=1/19, |a_(n+1)/a_n|=1/(n+1), 1/22, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
44. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":20,"start_index":20}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(20, 20, epsilon=1/20, |a_(n+1)/a_n|=1/(n+1), 1/21, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
45. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":21,"start_index":22}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(21, 22, epsilon=1/21, |a_(n+1)/a_n|=1/(n+1), 1/23, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
46. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":22,"start_index":24}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(22, 24, epsilon=1/22, |a_(n+1)/a_n|=1/(n+1), 1/25, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
47. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":23,"start_index":23}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(23, 23, epsilon=1/23, |a_(n+1)/a_n|=1/(n+1), 1/24, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
48. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":24,"start_index":25}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(24, 25, epsilon=1/24, |a_(n+1)/a_n|=1/(n+1), 1/26, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
49. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":25,"start_index":27}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(25, 27, epsilon=1/25, |a_(n+1)/a_n|=1/(n+1), 1/28, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
50. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-infinite-radius-certificate；資料={"epsilon_denominator":26,"start_index":26}。請依 [epsilon_denominator, N, epsilon, coefficient_ratio_law, ratio_bound_at_N, strict_tail_certificate, certificate] 欄序回答。
此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。
查看答案與解釋
(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)
Wave6第七層核對=(26, 26, epsilon=1/26, |a_(n+1)/a_n|=1/(n+1), 1/27, 是, for_all_n_at_least_N_coefficient_ratio_is_less_than_epsilon_so_the_ratio_limit_is_zero_and_radius_is_infinity)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-infinite-radius-certificate · candidate
51. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[1,2],"y":[-2,3],"degree":0}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(1/2, -2/3, 0, [[0, 1]], 1, 1, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(1/2, -2/3, 0, [[0, 1]], 1, 1, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
52. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[2,3],"y":[-3,4],"degree":1}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(2/3, -3/4, 1, [[0, -3/4], [1, 2/3]], -1/12, -1/12, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
53. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[3,4],"y":[-4,5],"degree":2}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(3/4, -4/5, 2, [[0, 8/25], [1, -3/5], [2, 9/32]], 1/800, 1/800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
54. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[4,5],"y":[-5,6],"degree":3}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(4/5, -5/6, 3, [[0, -125/1296], [1, 5/18], [2, -4/15], [3, 32/375]], -1/162000, -1/162000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
55. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[5,6],"y":[-6,7],"degree":4}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(5/6, -6/7, 4, [[0, 54/2401], [1, -30/343], [2, 25/196], [3, -125/1512], [4, 625/31104]], 1/74680704, 1/74680704, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
56. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[6,7],"y":[-7,8],"degree":5}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(6/7, -7/8, 5, [[0, -16807/3932160], [1, 343/16384], [2, -21/512], [3, 9/224], [4, -27/1372], [5, 324/84035]], -1/66087813120, -1/66087813120, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
57. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[7,8],"y":[-8,9],"degree":6}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(7/8, -8/9, 6, [[0, 16384/23914845], [1, -3584/885735], [2, 196/19683], [3, -343/26244], [4, 2401/248832], [5, -16807/4423680], [6, 117649/188743680]], 1/100306130042880, 1/100306130042880, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
58. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[8,9],"y":[-9,10],"degree":7}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(8/9, -9/10, 7, [[0, -531441/5600000000], [1, 6561/10000000], [2, -243/125000], [3, 2/625], [4, -32/10125], [5, 512/273375], [6, -8192/13286025], [7, 131072/1506635235]], -1/241061637600000000, -1/241061637600000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
59. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[9,10],"y":[-10,11],"degree":8}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(9/10, -10/11, 8, [[0, 156250/13504609503], [1, -12500/136410197], [2, 1125/3543122], [3, -405/644204], [4, 729/937024], [5, -6561/10648000], [6, 59049/193600000], [7, -531441/6160000000], [8, 4782969/448000000000]], 1/864295008192000000000, 1/864295008192000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
60. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[10,11],"y":[-11,12],"degree":9}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(10/11, -11/12, 9, [[0, -2357947691/1872381094133760], [1, 19487171/1733686198272], [2, -805255/18059231232], [3, 33275/322486272], [4, -1375/8957952], [5, 625/4105728], [6, -3125/31049568], [7, 15625/365263668], [8, -78125/7366150638], [9, 1562500/1336956340797]], -1/4414976677584753037148160, -1/4414976677584753037148160, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
61. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[11,12],"y":[-12,13],"degree":10}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(11/12, -12/13, 10, [[0, 2985984/24125236073575], [1, -456192/371157478055], [2, 156816/28550575235], [3, -31944/2196198095], [4, 14641/579217080], [5, -161051/5346619200], [6, 1771561/71068907520], [7, -19487171/1377643438080], [8, 214358881/40693467709440], [9, -2357947691/2028412851978240], [10, 25937424601/224685731296051200]], 1/30974836056463278561086668800, 1/30974836056463278561086668800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
62. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[12,13],"y":[-13,14],"degree":11}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(12/13, -13/14, 11, [[0, -1792160394037/161645682964443955200], [1, 10604499373/87470607664742400], [2, -62748517/104131675791360], [3, 371293/206610467840], [4, -6591/1844736320], [5, 117/23529800], [6, -27/5462275], [7, 648/184624895], [8, -7776/4457372465], [9, 62208/107613706655], [10, -1492992/12990511731925], [11, 35831808/3449908758521225]], -1/289694990875937857011834775142400, -1/289694990875937857011834775142400, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
63. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[13,14],"y":[-14,15],"degree":12}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(13/14, -14/15, 12, [[0, 7909306972/8670299029541015625], [1, -2098387564/192673311767578125], [2, 139178767/2335434082031250], [3, -36924979/186834726562500], [4, 9796423/22143375000000], [5, -2599051/3690562500000], [6, 4826809/5904900000000], [7, -62748517/90016920000000], [8, 815730721/1881953740800000], [9, -10604499373/55329439979520000], [10, 137858491849/2409904496885760000], [11, -1792160394037/173191803176189952000], [12, 23298085122481/27156474738026584473600]], 1/3523453147278219257625600000000000000, 1/3523453147278219257625600000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
64. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[14,15],"y":[-15,16],"degree":13}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(14/15, -15/16, 13, [[0, -320361328125/4616297704445815291904], [1, 1423828125/1585267068834414592], [2, -132890625/24769797950537728], [3, 1378125/70368744177664], [4, -214375/4398046511104], [5, 12005/137438953472], [6, -16807/144955146240], [7, 117649/1019215872000], [8, -823543/9555148800000], [9, 5764801/120932352000000], [10, -40353607/2125764000000000], [11, 282475249/54804853125000000], [12, -1977326743/2312079741210937500], [13, 110730297608/1690708310760498046875]], -1/54579111145812232494907392000000000000000, -1/54579111145812232494907392000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
65. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[15,16],"y":[-16,17],"degree":14}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(15/16, -16/17, 14, [[0, 35184372088832/7167427341512963030230725], [1, -274877906944/4015365457430231389485], [2, 536870912/1211271631200673119], [3, -41943040/23750424141189669], [4, 204800/42335871909429], [5, -8000/830115135479], [6, 5625/390642416696], [7, -84375/5147288314112], [8, 1265625/88585264431104], [9, -6328125/666994932187136], [10, 94921875/20088317957636096], [11, -1423828125/831892696598577152], [12, 21357421875/50109301253937823744], [13, -320361328125/4904816310973678747648], [14, 4805419921875/1034050685795862625386496]], 1/1057725582686384458266478824154331142212812800, 1/1057725582686384458266478824154331142212812800, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
66. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[16,17],"y":[-17,18],"degree":15}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(16/17, -17/18, 15, [[0, -2862423051509815793/8822409004275290841311870976000], [1, 9904578032905937/2042224306545206213266636800], [2, -34271896307633/1013008088564090383564800], [3, 118587876497/811705199169944217600], [4, -410338673/939473610150398400], [5, 1419857/1482755066525250], [6, -78608/49425168884175], [7, 8704/4271310891225], [8, -16384/8068031683425], [9, 524288/333094450929975], [10, -16777216/17826721540511625], [11, 536870912/1259358839495254575], [12, -17179869184/121318234871376190725], [13, 549755813888/16881207718954086983475], [14, -17592186044416/3794520357271568663063325], [15, 562949953421312/1827693972085805572708834875]], -1/25253466903685353496585230352209316437943123968000, -1/25253466903685353496585230352209316437943123968000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
67. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[17,18],"y":[-18,19],"degree":16}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(17/18, -18/19, 16, [[0, 5083731656658/252638623108540190242595875], [1, -4267824106824/13296769637291588960136625], [2, 335893563963/139965996182016725896175], [3, -23498726697/2104751822285965802950], [4, 4931831529/136340199014475517600], [5, -3105227259/35878999740651452000], [6, 217238121/1373358841747424000], [7, -410338673/1821507516422899200], [8, 6975757441/27610219196304998400], [9, -118587876497/529680257739640627200], [10, 2015993900449/12903488985537110016000], [11, -34271896307633/403403813547844386816000], [12, 582622237229761/16509831863937041852006400], [13, -9904578032905937/914992260669773951061196800], [14, 168377826559400929/72814120954352537579185766400], [15, -2862423051509815793/9312542837846140332495863808000], [16, 48661191875666868481/2540853793231283762297818841088000]], 1/732887459788283721221778416595587951601772740476928000, 1/732887459788283721221778416595587951601772740476928000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
68. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[18,19],"y":[-19,20],"degree":17}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(18/19, -19/20, 17, [[0, -5480386857784802185939/4662066257539891200000000000000000000], [1, 15181127029874798299/761775532277760000000000000000000], [2, -42052983462257059/264505393152000000000000000000], [3, 116490258898219/146947440640000000000000000], [4, -2904189280011/1049624576000000000000000], [5, 72403610859/10092544000000000000000], [6, -1805076171/126156800000000000000], [7, 45001899/2007040000000000000], [8, -1121931/40140800000000000], [9, 531441/19066880000000000], [10, -4782969/215098240000000000], [11, 43046721/3050553968000000000], [12, -387420489/55062499122400000000], [13, 3486784401/1292041541907116000000], [14, -31381059609/40812362204991026650000], [15, 282429536481/1841657844500220077581250], [16, -2541865828329/132967696372915889601366250], [17, 91507169819844/81602275264058481448358467625]], -1/25549926647943596675599717954252229836800000000000000000000, -1/25549926647943596675599717954252229836800000000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
69. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[19,20],"y":[-20,21],"degree":18}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(19/20, -20/21, 18, [[0, 32000000000000000/493058422447458087543290824655799], [1, -3040000000000000/2608774721944222685414237167491], [2, 72200000000000/7307492218331155981552485063], [3, -54872000000000/1043927459761593711650355009], [4, 3258025000000/16570277139072916057942143], [5, -61902475000/112722973735189905156069], [6, 5880735125/4954855988359995831036], [7, -4469358695/2202158217048887036016], [8, 16983563041/6101217571044535511040], [9, -322687697779/104592301217906323046400], [10, 6131066257801/2213593676569445990400000], [11, -116490258898219/57975072481580728320000000], [12, 2213314919066161/1893063591235289088000000000], [13, -42052983462257059/78126433923996057600000000000], [14, 799006685782884121/4166743142613123072000000000000], [15, -15181127029874798299/297624510186651648000000000000000], [16, 288441413567621167681/30234870876104294400000000000000000], [17, -5480386857784802185939/4895169570416885760000000000000000000], [18, 104127350297911241532841/1678343852714360832000000000000000000000]], 1/1058834899714592382164569538056763580874752000000000000000000000, 1/1058834899714592382164569538056763580874752000000000000000000000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
70. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[20,21],"y":[-21,22],"degree":19}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(20/21, -21/22, 19, [[0, -41209797661291758429/12132755312575326542353091425992704000], [1, 93446253200208069/1451286520642981643822140122726400], [2, -1907066391840981/3298378456006776463232136642560], [3, 1441471195647/440959686631921987063119872], [4, -16343210835/1252726382477051099611136], [5, 555891525/14235527073602853404672], [6, -14706125/161767353109123334144], [7, 214375/1272645260473872384], [8, -21875/86771267759582208], [9, 15625/50825732251738752], [10, -78125/254706226397918064], [11, 390625/1560075636687248142], [12, -15625000/93817275788055876903], [13, 625000000/6985122624583433016687], [14, -25000000000/653426470972395688560993], [15, 1000000000000/78589383736043590542381249], [16, -10000000000000/3150719838872293039017284619], [17, 400000000000000/715786261576532754954926751171], [18, -16000000000000000/258268697472478045856009479581609], [19, 640000000000000000/196730310556535776929773039037663801]], -1/51676388762462995371436759840422682365878499503552553575761575936000, -1/51676388762462995371436759840422682365878499503552553575761575936000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
71. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[21,22],"y":[-22,23],"degree":20}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(21/22, -22/23, 20, [[0, 244636361793658185164/1447933879643515222582682716129837194375], [1, -2021788113997175084/599558542295451437922435907300139625], [2, 4177248169415651/130665477235578388999114287305250], [3, -241658985007517/1262468379087713903373084901500], [4, 13980271859939/17220370047232244206282488000], [5, -808776058013/311963225493337757360190000], [6, 46788697571/7233929866512179880816000], [7, -2706784157/209679126565570431328000], [8, 469772457/22440575418388808704000], [9, -27176919/975677192103861248000], [10, 51883209/1696829899311063040000], [11, -1089547389/39277923408400433152000], [12, 68641485507/3306176509507097329664000], [13, -1441471195647/113056861944666610860032000], [14, 30270895108587/4758219233149446926630912000], [15, -635688797280327/250323707483079599183626240000], [16, 13349464742886867/16856580789121464487634796544000], [17, -280338759600624207/1507564638400993584828903325696000], [18, 1962371317204369449/63448807390093990874538192142336000], [19, -41209797661291758429/12684244190419659567005504672628736000], [20, 865405750887126927009/5338412337533143678635360227436789760000]], 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 1/2945336732372916073844860985203962504337568772809813725658735066480640000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
72. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[22,23],"y":[-23,24],"degree":0}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(22/23, -23/24, 0, [[0, 1]], 1, 1, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(22/23, -23/24, 0, [[0, 1]], 1, 1, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
73. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[23,24],"y":[-24,25],"degree":1}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(23/24, -24/25, 1, [[0, -24/25], [1, 23/24]], -1/600, -1/600, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
74. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[24,25],"y":[-25,26],"degree":2}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(24/25, -25/26, 2, [[0, 625/1352], [1, -12/13], [2, 288/625]], 1/845000, 1/845000, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
75. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-addition-convolution；資料={"x":[25,26],"y":[-26,27],"degree":3}。請依 [x, y, degree, coefficient_terms, convolution_coefficient, sum_argument_coefficient, equal, certificate] 欄序回答。
查看答案與解釋
(25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)
Wave6第七層核對=(25/26, -26/27, 3, [[0, -8788/59049], [1, 325/729], [2, -625/1404], [3, 15625/105456]], -1/2075690448, -1/2075690448, 是, binomial_convolution_matches_the_declared_degree;the_general_addition_law_requires_all_degrees_plus_absolute_convergence_for_Cauchy_product_rearrangement)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-addition-convolution · candidate
76. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-12}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-12, E(0)=1, E(-12)*E(12)=1, E(-6)*E(6)=1, E(-6)!=0, E(-12)=E(-6)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
77. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-11}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-11, E(0)=1, E(-11)*E(11)=1, E(-11/2)*E(11/2)=1, E(-11/2)!=0, E(-11)=E(-11/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
78. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-10}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-10, E(0)=1, E(-10)*E(10)=1, E(-5)*E(5)=1, E(-5)!=0, E(-10)=E(-5)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
79. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-9}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-9, E(0)=1, E(-9)*E(9)=1, E(-9/2)*E(9/2)=1, E(-9/2)!=0, E(-9)=E(-9/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
80. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-8}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-8, E(0)=1, E(-8)*E(8)=1, E(-4)*E(4)=1, E(-4)!=0, E(-8)=E(-4)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
81. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-7}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-7, E(0)=1, E(-7)*E(7)=1, E(-7/2)*E(7/2)=1, E(-7/2)!=0, E(-7)=E(-7/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
82. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-6}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-6, E(0)=1, E(-6)*E(6)=1, E(-3)*E(3)=1, E(-3)!=0, E(-6)=E(-3)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
83. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-5}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-5, E(0)=1, E(-5)*E(5)=1, E(-5/2)*E(5/2)=1, E(-5/2)!=0, E(-5)=E(-5/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
84. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-4}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-4, E(0)=1, E(-4)*E(4)=1, E(-2)*E(2)=1, E(-2)!=0, E(-4)=E(-2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
85. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-3}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-3, E(0)=1, E(-3)*E(3)=1, E(-3/2)*E(3/2)=1, E(-3/2)!=0, E(-3)=E(-3/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
86. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-2}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-2, E(0)=1, E(-2)*E(2)=1, E(-1)*E(1)=1, E(-1)!=0, E(-2)=E(-1)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
87. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":-1}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(-1, E(0)=1, E(-1)*E(1)=1, E(-1/2)*E(1/2)=1, E(-1/2)!=0, E(-1)=E(-1/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
88. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":0}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(0, E(0)=1, E(0)*E(0)=1, E(0)*E(0)=1, E(0)!=0, E(0)=E(0)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
89. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":1}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(1, E(0)=1, E(1)*E(-1)=1, E(1/2)*E(-1/2)=1, E(1/2)!=0, E(1)=E(1/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
90. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":2}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(2, E(0)=1, E(2)*E(-2)=1, E(1)*E(-1)=1, E(1)!=0, E(2)=E(1)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
91. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":3}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(3, E(0)=1, E(3)*E(-3)=1, E(3/2)*E(-3/2)=1, E(3/2)!=0, E(3)=E(3/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
92. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":4}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(4, E(0)=1, E(4)*E(-4)=1, E(2)*E(-2)=1, E(2)!=0, E(4)=E(2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
93. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":5}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(5, E(0)=1, E(5)*E(-5)=1, E(5/2)*E(-5/2)=1, E(5/2)!=0, E(5)=E(5/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
94. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":6}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(6, E(0)=1, E(6)*E(-6)=1, E(3)*E(-3)=1, E(3)!=0, E(6)=E(3)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
95. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":7}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(7, E(0)=1, E(7)*E(-7)=1, E(7/2)*E(-7/2)=1, E(7/2)!=0, E(7)=E(7/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
96. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":8}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(8, E(0)=1, E(8)*E(-8)=1, E(4)*E(-4)=1, E(4)!=0, E(8)=E(4)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
97. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":9}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(9, E(0)=1, E(9)*E(-9)=1, E(9/2)*E(-9/2)=1, E(9/2)!=0, E(9)=E(9/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
98. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":10}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(10, E(0)=1, E(10)*E(-10)=1, E(5)*E(-5)=1, E(5)!=0, E(10)=E(5)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
99. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":11}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(11, E(0)=1, E(11)*E(-11)=1, E(11/2)*E(-11/2)=1, E(11/2)!=0, E(11)=E(11/2)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
100. 實指數函數
E(x)=sum_n>=0 x^n/n!的完整coefficient law與exact rational certificate。操作=exp-zero-inverse-positivity；資料={"x":12}。請依 [x, zero_identity, inverse_identity, half_inverse_identity, half_nonzero, strict_square_identity, symbolic_positivity_certificate, certificate] 欄序回答。
查看答案與解釋
(12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)
Wave6第七層核對=(12, E(0)=1, E(12)*E(-12)=1, E(6)*E(-6)=1, E(6)!=0, E(12)=E(6)^2>0, 是, addition_law_gives_inverse_nonzero_half_value_and_strict_square_positivity_without_claiming_a_rational_exact_value)；依完整明示資料獨立重算；finite partial sum不是E(x)；coefficient-ratio limit給radius infinity；單一degree只核該degree，一般addition law需所有degree的binomial identity與absolute convergence；E(x/2)E(-x/2)=1先給half-value非零，再由平方推出嚴格正；不外推derivative、log、ODE、injectivity或非零有理exact value。
本題逐步詳解（AI 二次覆核）
實指數函數 · qf-mod44-exp-zero-inverse-positivity · candidate
原始題目 JSONL · 驗收報告