# 冪級數收斂半徑基礎 · 脈衝 R05

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-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations · 2026-10-14 · 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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[1,2],"center":-12}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0001/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 2. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[2,3],"center":-11}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0002/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 3. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[3,4],"center":-10}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0003/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 4. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[4,5],"center":-9}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0004/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 5. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[5,6],"center":-8}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0005/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 6. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[6,7],"center":-7}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0006/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 7. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[7,8],"center":-6}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0007/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 8. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[8,9],"center":-5}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0008/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 9. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[9,10],"center":-4}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0009/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 10. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[10,11],"center":-3}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0010/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 11. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[11,12],"center":-2}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0011/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 12. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[12,13],"center":-1}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0012/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 13. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[13,14],"center":0}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0013/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 14. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[14,15],"center":1}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0014/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 15. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[15,16],"center":2}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0015/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 16. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[16,17],"center":3}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0016/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 17. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[17,18],"center":4}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0017/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 18. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[18,19],"center":5}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0018/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 19. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[19,20],"center":6}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0019/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 20. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[20,21],"center":7}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0020/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 21. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[21,22],"center":8}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0021/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 22. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[22,23],"center":9}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0022/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 23. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[23,24],"center":10}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0023/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 24. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[24,25],"center":11}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0024/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 25. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-geometric-exact；資料={"coefficient_ratio":[25,26],"center":12}。請依 [coefficient_law, center, radius, interior_status, exterior_status, boundary] 欄序回答。

查看答案與解釋(a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)

Wave6第六層核對=(a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0025/)

冪級數收斂半徑 · qf-mod43-radius-geometric-exact · candidate





## 26. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[1,2],"center":-12,"point":-12}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0026/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 27. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[2,3],"center":-11,"point":-9}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 28. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[3,4],"center":-10,"point":-10}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0028/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 29. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[4,5],"center":-9,"point":-7}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0029/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 30. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[5,6],"center":-8,"point":-8}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0030/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 31. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[6,7],"center":-7,"point":-5}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0031/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 32. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[7,8],"center":-6,"point":-6}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0032/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 33. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[8,9],"center":-5,"point":-3}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0033/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 34. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[9,10],"center":-4,"point":-4}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0034/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 35. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[10,11],"center":-3,"point":-1}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0035/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 36. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[11,12],"center":-2,"point":-2}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0036/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 37. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[12,13],"center":-1,"point":1}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0037/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 38. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[13,14],"center":0,"point":0}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0038/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 39. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[14,15],"center":1,"point":3}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0039/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 40. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[15,16],"center":2,"point":2}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0040/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 41. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[16,17],"center":3,"point":5}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0041/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 42. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[17,18],"center":4,"point":4}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0042/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 43. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[18,19],"center":5,"point":7}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0043/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 44. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[19,20],"center":6,"point":6}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0044/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 45. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[20,21],"center":7,"point":9}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0045/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 46. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[21,22],"center":8,"point":8}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0046/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 47. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[22,23],"center":9,"point":11}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0047/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 48. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[23,24],"center":10,"point":10}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0048/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 49. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[24,25],"center":11,"point":13}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, 否, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, 否, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0049/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 50. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-inside-outside-audit；資料={"coefficient_ratio":[25,26],"center":12,"point":12}。請依 [coefficient_law, center, point, distance, radius, classification, absolutely_convergent, certificate] 欄序回答。

查看答案與解釋(a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, 是, classification_uses_exact_distance_from_center)

Wave6第六層核對=(a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, 是, classification_uses_exact_distance_from_center)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0050/)

冪級數收斂半徑 · qf-mod43-radius-inside-outside-audit · candidate





## 51. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-12,"radius":1,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=1^(-n)/(n+1)^0_for_every_n>=0, -12, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0051/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 52. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-11,"radius":2,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=2^(-n)/(n+1)^1_for_every_n>=0, -11, 2, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0052/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 53. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-10,"radius":3,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=3^(-n)/(n+1)^2_for_every_n>=0, -10, 3, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0053/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 54. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-9,"radius":4,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=4^(-n)/(n+1)^0_for_every_n>=0, -9, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0054/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 55. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-8,"radius":5,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=5^(-n)/(n+1)^1_for_every_n>=0, -8, 5, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0055/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 56. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-7,"radius":1,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=1^(-n)/(n+1)^2_for_every_n>=0, -7, 1, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0056/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 57. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-6,"radius":2,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=2^(-n)/(n+1)^0_for_every_n>=0, -6, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0057/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 58. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-5,"radius":3,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=3^(-n)/(n+1)^1_for_every_n>=0, -5, 3, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0058/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 59. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-4,"radius":4,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=4^(-n)/(n+1)^2_for_every_n>=0, -4, 4, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0059/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 60. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-3,"radius":5,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=5^(-n)/(n+1)^0_for_every_n>=0, -3, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0060/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 61. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-2,"radius":1,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=1^(-n)/(n+1)^1_for_every_n>=0, -2, 1, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0061/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 62. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":-1,"radius":2,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=2^(-n)/(n+1)^2_for_every_n>=0, -1, 2, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0062/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 63. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":0,"radius":3,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=3^(-n)/(n+1)^0_for_every_n>=0, 0, 3, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0063/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 64. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":1,"radius":4,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=4^(-n)/(n+1)^1_for_every_n>=0, 1, 4, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0064/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 65. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":2,"radius":5,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=5^(-n)/(n+1)^2_for_every_n>=0, 2, 5, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0065/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 66. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":3,"radius":1,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=1^(-n)/(n+1)^0_for_every_n>=0, 3, 1, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0066/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 67. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":4,"radius":2,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=2^(-n)/(n+1)^1_for_every_n>=0, 4, 2, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0067/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 68. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":5,"radius":3,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=3^(-n)/(n+1)^2_for_every_n>=0, 5, 3, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0068/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 69. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":6,"radius":4,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=4^(-n)/(n+1)^0_for_every_n>=0, 6, 4, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0069/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 70. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":7,"radius":5,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=5^(-n)/(n+1)^1_for_every_n>=0, 7, 5, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0070/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 71. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":8,"radius":1,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=1^(-n)/(n+1)^2_for_every_n>=0, 8, 1, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0071/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 72. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":9,"radius":2,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=2^(-n)/(n+1)^0_for_every_n>=0, 9, 2, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0072/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 73. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":10,"radius":3,"decay_power":1}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=3^(-n)/(n+1)^1_for_every_n>=0, 10, 3, 1, divergence, conditional_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0073/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 74. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":11,"radius":4,"decay_power":2}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=4^(-n)/(n+1)^2_for_every_n>=0, 11, 4, 2, absolute_convergence, absolute_convergence, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0074/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 75. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={"center":12,"radius":5,"decay_power":0}。請依 [coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary] 欄序回答。

positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。

查看答案與解釋(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)

Wave6第六層核對=(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075/)

冪級數收斂半徑 · qf-mod43-radius-endpoint-independent · candidate





## 76. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":1,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0076/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 77. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":2,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0077/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 78. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":3,"prefix_length":1,"first_ratio":[1,2],"second_ratio":[1,3]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 3, 1, [0, 0], a_n=0_for_n<=1;a_n=3*(1/2)^(n-2)_for_n>1, b_n=0_for_n<=1;b_n=3*(1/3)^(n-2)_for_n>1, 2, 3, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0078/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 79. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":4,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0079/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 80. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":5,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0080/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 81. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":6,"prefix_length":2,"first_ratio":[1,2],"second_ratio":[1,4]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 6, 2, [0, 0, 0], a_n=0_for_n<=2;a_n=6*(1/2)^(n-3)_for_n>2, b_n=0_for_n<=2;b_n=6*(1/4)^(n-3)_for_n>2, 2, 4, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 82. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":7,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0082/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 83. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":8,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0083/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 84. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":9,"prefix_length":3,"first_ratio":[1,2],"second_ratio":[1,5]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 9, 3, [0, 0, 0, 0], a_n=0_for_n<=3;a_n=9*(1/2)^(n-4)_for_n>3, b_n=0_for_n<=3;b_n=9*(1/5)^(n-4)_for_n>3, 2, 5, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0084/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 85. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":10,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0085/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 86. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":11,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0086/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 87. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":12,"prefix_length":4,"first_ratio":[1,2],"second_ratio":[1,6]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 88. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":13,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0088/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 89. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":14,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0089/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 90. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":15,"prefix_length":5,"first_ratio":[1,2],"second_ratio":[1,7]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 15, 5, [0, 0, 0, 0, 0, 0], a_n=0_for_n<=5;a_n=15*(1/2)^(n-6)_for_n>5, b_n=0_for_n<=5;b_n=15*(1/7)^(n-6)_for_n>5, 2, 7, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0090/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 91. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":16,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0091/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 92. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":17,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0092/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 93. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":18,"prefix_length":6,"first_ratio":[1,2],"second_ratio":[1,8]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 18, 6, [0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=6;a_n=18*(1/2)^(n-7)_for_n>6, b_n=0_for_n<=6;b_n=18*(1/8)^(n-7)_for_n>6, 2, 8, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0093/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 94. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":19,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0094/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 95. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":20,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0095/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 96. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":21,"prefix_length":7,"first_ratio":[1,2],"second_ratio":[1,9]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 21, 7, [0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=7;a_n=21*(1/2)^(n-8)_for_n>7, b_n=0_for_n<=7;b_n=21*(1/9)^(n-8)_for_n>7, 2, 9, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0096/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 97. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":22,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0097/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 98. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"infinite_radius","scale":23,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)

Wave6第六層核對=(infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0098/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 99. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"finite_prefix","scale":24,"prefix_length":8,"first_ratio":[1,2],"second_ratio":[1,10]}。請依 [mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)

Wave6第六層核對=(finite_prefix, 24, 8, [0, 0, 0, 0, 0, 0, 0, 0, 0], a_n=0_for_n<=8;a_n=24*(1/2)^(n-9)_for_n>8, b_n=0_for_n<=8;b_n=24*(1/10)^(n-9)_for_n>8, 2, 10, 是, 是, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0099/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate





## 100. 冪級數收斂半徑

明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-zero-infinity-prefix-boundary；資料={"mode":"zero_radius","scale":25,"prefix_length":0,"first_ratio":1,"second_ratio":1}。請依 [mode, scale, coefficient_law, radius, symbolic_certificate, certificate] 欄序回答。

mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。

查看答案與解釋(zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)

Wave6第六層核對=(zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0100/)

冪級數收斂半徑 · qf-mod43-radius-zero-infinity-prefix-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/validation-report.json)