{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0001","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(1/2)^n_for_every_n>=0, -12, 2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125001,"parameters":{"coefficient_ratio":[1,2],"center":-12,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-12,\"coefficient_ratio\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0002","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(2/3)^n_for_every_n>=0, -11, 3/2, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125002,"parameters":{"coefficient_ratio":[2,3],"center":-11,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-11,\"coefficient_ratio\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0003","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(3/4)^n_for_every_n>=0, -10, 4/3, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125003,"parameters":{"coefficient_ratio":[3,4],"center":-10,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-10,\"coefficient_ratio\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0004","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(4/5)^n_for_every_n>=0, -9, 5/4, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125004,"parameters":{"coefficient_ratio":[4,5],"center":-9,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-9,\"coefficient_ratio\":[4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0005","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(5/6)^n_for_every_n>=0, -8, 6/5, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125005,"parameters":{"coefficient_ratio":[5,6],"center":-8,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-8,\"coefficient_ratio\":[5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0006","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(6/7)^n_for_every_n>=0, -7, 7/6, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125006,"parameters":{"coefficient_ratio":[6,7],"center":-7,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-7,\"coefficient_ratio\":[6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0007","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(7/8)^n_for_every_n>=0, -6, 8/7, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125007,"parameters":{"coefficient_ratio":[7,8],"center":-6,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-6,\"coefficient_ratio\":[7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0008","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(8/9)^n_for_every_n>=0, -5, 9/8, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125008,"parameters":{"coefficient_ratio":[8,9],"center":-5,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-5,\"coefficient_ratio\":[8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0009","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(9/10)^n_for_every_n>=0, -4, 10/9, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125009,"parameters":{"coefficient_ratio":[9,10],"center":-4,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-4,\"coefficient_ratio\":[9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0010","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(10/11)^n_for_every_n>=0, -3, 11/10, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125010,"parameters":{"coefficient_ratio":[10,11],"center":-3,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-3,\"coefficient_ratio\":[10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0011","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(11/12)^n_for_every_n>=0, -2, 12/11, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125011,"parameters":{"coefficient_ratio":[11,12],"center":-2,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-2,\"coefficient_ratio\":[11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0012","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(12/13)^n_for_every_n>=0, -1, 13/12, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125012,"parameters":{"coefficient_ratio":[12,13],"center":-1,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":-1,\"coefficient_ratio\":[12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0013","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(13/14)^n_for_every_n>=0, 0, 14/13, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125013,"parameters":{"coefficient_ratio":[13,14],"center":0,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":0,\"coefficient_ratio\":[13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0014","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(14/15)^n_for_every_n>=0, 1, 15/14, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125014,"parameters":{"coefficient_ratio":[14,15],"center":1,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":1,\"coefficient_ratio\":[14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0015","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(15/16)^n_for_every_n>=0, 2, 16/15, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125015,"parameters":{"coefficient_ratio":[15,16],"center":2,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":2,\"coefficient_ratio\":[15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0016","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(16/17)^n_for_every_n>=0, 3, 17/16, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125016,"parameters":{"coefficient_ratio":[16,17],"center":3,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":3,\"coefficient_ratio\":[16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0017","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(17/18)^n_for_every_n>=0, 4, 18/17, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125017,"parameters":{"coefficient_ratio":[17,18],"center":4,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":4,\"coefficient_ratio\":[17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0018","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(18/19)^n_for_every_n>=0, 5, 19/18, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125018,"parameters":{"coefficient_ratio":[18,19],"center":5,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":5,\"coefficient_ratio\":[18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0019","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(19/20)^n_for_every_n>=0, 6, 20/19, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125019,"parameters":{"coefficient_ratio":[19,20],"center":6,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":6,\"coefficient_ratio\":[19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0020","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(20/21)^n_for_every_n>=0, 7, 21/20, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125020,"parameters":{"coefficient_ratio":[20,21],"center":7,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":7,\"coefficient_ratio\":[20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0021","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(21/22)^n_for_every_n>=0, 8, 22/21, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125021,"parameters":{"coefficient_ratio":[21,22],"center":8,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":8,\"coefficient_ratio\":[21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0022","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(22/23)^n_for_every_n>=0, 9, 23/22, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125022,"parameters":{"coefficient_ratio":[22,23],"center":9,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":9,\"coefficient_ratio\":[22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0023","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(23/24)^n_for_every_n>=0, 10, 24/23, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125023,"parameters":{"coefficient_ratio":[23,24],"center":10,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":10,\"coefficient_ratio\":[23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0024","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(24/25)^n_for_every_n>=0, 11, 25/24, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125024,"parameters":{"coefficient_ratio":[24,25],"center":11,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":11,\"coefficient_ratio\":[24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0025","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(25/26)^n_for_every_n>=0, 12, 26/25, absolute_convergence_inside, divergence_outside, endpoints_require_separate_audit)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-geometric-exact","seed":202611125025,"parameters":{"coefficient_ratio":[25,26],"center":12,"domain":"Exact_power_series_radius","operation":"radius-geometric-exact","semantic_template":"qf-mod43-radius-geometric-exact/{\"center\":12,\"coefficient_ratio\":[25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0026","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(1/2)^n_for_every_n>=0, -12, -12, 0, 2, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125026,"parameters":{"coefficient_ratio":[1,2],"center":-12,"point":-12,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-12,\"coefficient_ratio\":[1,2],\"point\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0027","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(2/3)^n_for_every_n>=0, -11, -9, 2, 3/2, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125027,"parameters":{"coefficient_ratio":[2,3],"center":-11,"point":-9,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-11,\"coefficient_ratio\":[2,3],\"point\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0028","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(3/4)^n_for_every_n>=0, -10, -10, 0, 4/3, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125028,"parameters":{"coefficient_ratio":[3,4],"center":-10,"point":-10,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-10,\"coefficient_ratio\":[3,4],\"point\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0029","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(4/5)^n_for_every_n>=0, -9, -7, 2, 5/4, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125029,"parameters":{"coefficient_ratio":[4,5],"center":-9,"point":-7,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-9,\"coefficient_ratio\":[4,5],\"point\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0030","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(5/6)^n_for_every_n>=0, -8, -8, 0, 6/5, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125030,"parameters":{"coefficient_ratio":[5,6],"center":-8,"point":-8,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-8,\"coefficient_ratio\":[5,6],\"point\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0031","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(6/7)^n_for_every_n>=0, -7, -5, 2, 7/6, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125031,"parameters":{"coefficient_ratio":[6,7],"center":-7,"point":-5,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-7,\"coefficient_ratio\":[6,7],\"point\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0032","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(7/8)^n_for_every_n>=0, -6, -6, 0, 8/7, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125032,"parameters":{"coefficient_ratio":[7,8],"center":-6,"point":-6,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-6,\"coefficient_ratio\":[7,8],\"point\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0033","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(8/9)^n_for_every_n>=0, -5, -3, 2, 9/8, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125033,"parameters":{"coefficient_ratio":[8,9],"center":-5,"point":-3,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-5,\"coefficient_ratio\":[8,9],\"point\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0034","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(9/10)^n_for_every_n>=0, -4, -4, 0, 10/9, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125034,"parameters":{"coefficient_ratio":[9,10],"center":-4,"point":-4,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-4,\"coefficient_ratio\":[9,10],\"point\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0035","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(10/11)^n_for_every_n>=0, -3, -1, 2, 11/10, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125035,"parameters":{"coefficient_ratio":[10,11],"center":-3,"point":-1,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-3,\"coefficient_ratio\":[10,11],\"point\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0036","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(11/12)^n_for_every_n>=0, -2, -2, 0, 12/11, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125036,"parameters":{"coefficient_ratio":[11,12],"center":-2,"point":-2,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-2,\"coefficient_ratio\":[11,12],\"point\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0037","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(12/13)^n_for_every_n>=0, -1, 1, 2, 13/12, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125037,"parameters":{"coefficient_ratio":[12,13],"center":-1,"point":1,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":-1,\"coefficient_ratio\":[12,13],\"point\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0038","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(13/14)^n_for_every_n>=0, 0, 0, 0, 14/13, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125038,"parameters":{"coefficient_ratio":[13,14],"center":0,"point":0,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":0,\"coefficient_ratio\":[13,14],\"point\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0039","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(14/15)^n_for_every_n>=0, 1, 3, 2, 15/14, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125039,"parameters":{"coefficient_ratio":[14,15],"center":1,"point":3,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":1,\"coefficient_ratio\":[14,15],\"point\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0040","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(15/16)^n_for_every_n>=0, 2, 2, 0, 16/15, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125040,"parameters":{"coefficient_ratio":[15,16],"center":2,"point":2,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":2,\"coefficient_ratio\":[15,16],\"point\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0041","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(16/17)^n_for_every_n>=0, 3, 5, 2, 17/16, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125041,"parameters":{"coefficient_ratio":[16,17],"center":3,"point":5,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":3,\"coefficient_ratio\":[16,17],\"point\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0042","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(17/18)^n_for_every_n>=0, 4, 4, 0, 18/17, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125042,"parameters":{"coefficient_ratio":[17,18],"center":4,"point":4,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":4,\"coefficient_ratio\":[17,18],\"point\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0043","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(18/19)^n_for_every_n>=0, 5, 7, 2, 19/18, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125043,"parameters":{"coefficient_ratio":[18,19],"center":5,"point":7,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":5,\"coefficient_ratio\":[18,19],\"point\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0044","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(19/20)^n_for_every_n>=0, 6, 6, 0, 20/19, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125044,"parameters":{"coefficient_ratio":[19,20],"center":6,"point":6,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":6,\"coefficient_ratio\":[19,20],\"point\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0045","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(20/21)^n_for_every_n>=0, 7, 9, 2, 21/20, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125045,"parameters":{"coefficient_ratio":[20,21],"center":7,"point":9,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":7,\"coefficient_ratio\":[20,21],\"point\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0046","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(21/22)^n_for_every_n>=0, 8, 8, 0, 22/21, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125046,"parameters":{"coefficient_ratio":[21,22],"center":8,"point":8,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":8,\"coefficient_ratio\":[21,22],\"point\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0047","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(22/23)^n_for_every_n>=0, 9, 11, 2, 23/22, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125047,"parameters":{"coefficient_ratio":[22,23],"center":9,"point":11,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":9,\"coefficient_ratio\":[22,23],\"point\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0048","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(23/24)^n_for_every_n>=0, 10, 10, 0, 24/23, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125048,"parameters":{"coefficient_ratio":[23,24],"center":10,"point":10,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":10,\"coefficient_ratio\":[23,24],\"point\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0049","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(24/25)^n_for_every_n>=0, 11, 13, 2, 25/24, divergence, 否, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125049,"parameters":{"coefficient_ratio":[24,25],"center":11,"point":13,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":11,\"coefficient_ratio\":[24,25],\"point\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0050","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a_n=(25/26)^n_for_every_n>=0, 12, 12, 0, 26/25, absolute_convergence, 是, classification_uses_exact_distance_from_center)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-inside-outside-audit","seed":202611125050,"parameters":{"coefficient_ratio":[25,26],"center":12,"point":12,"domain":"Exact_power_series_radius","operation":"radius-inside-outside-audit","semantic_template":"qf-mod43-radius-inside-outside-audit/{\"center\":12,\"coefficient_ratio\":[25,26],\"point\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0051","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125051,"parameters":{"center":-12,"radius":1,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-12,\"decay_power\":0,\"radius\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0052","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125052,"parameters":{"center":-11,"radius":2,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-11,\"decay_power\":1,\"radius\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0053","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125053,"parameters":{"center":-10,"radius":3,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-10,\"decay_power\":2,\"radius\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0054","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125054,"parameters":{"center":-9,"radius":4,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-9,\"decay_power\":0,\"radius\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0055","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125055,"parameters":{"center":-8,"radius":5,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-8,\"decay_power\":1,\"radius\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0056","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125056,"parameters":{"center":-7,"radius":1,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-7,\"decay_power\":2,\"radius\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0057","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125057,"parameters":{"center":-6,"radius":2,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-6,\"decay_power\":0,\"radius\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0058","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125058,"parameters":{"center":-5,"radius":3,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-5,\"decay_power\":1,\"radius\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0059","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125059,"parameters":{"center":-4,"radius":4,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-4,\"decay_power\":2,\"radius\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0060","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125060,"parameters":{"center":-3,"radius":5,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-3,\"decay_power\":0,\"radius\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0061","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125061,"parameters":{"center":-2,"radius":1,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-2,\"decay_power\":1,\"radius\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0062","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125062,"parameters":{"center":-1,"radius":2,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":-1,\"decay_power\":2,\"radius\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0063","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125063,"parameters":{"center":0,"radius":3,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":0,\"decay_power\":0,\"radius\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0064","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125064,"parameters":{"center":1,"radius":4,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":1,\"decay_power\":1,\"radius\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0065","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125065,"parameters":{"center":2,"radius":5,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":2,\"decay_power\":2,\"radius\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0066","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125066,"parameters":{"center":3,"radius":1,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":3,\"decay_power\":0,\"radius\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0067","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125067,"parameters":{"center":4,"radius":2,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":4,\"decay_power\":1,\"radius\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0068","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125068,"parameters":{"center":5,"radius":3,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":5,\"decay_power\":2,\"radius\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0069","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125069,"parameters":{"center":6,"radius":4,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":6,\"decay_power\":0,\"radius\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0070","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125070,"parameters":{"center":7,"radius":5,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":7,\"decay_power\":1,\"radius\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0071","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125071,"parameters":{"center":8,"radius":1,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":8,\"decay_power\":2,\"radius\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0072","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125072,"parameters":{"center":9,"radius":2,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":9,\"decay_power\":0,\"radius\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0073","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125073,"parameters":{"center":10,"radius":3,"decay_power":1,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":10,\"decay_power\":1,\"radius\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0074","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125074,"parameters":{"center":11,"radius":4,"decay_power":2,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":11,\"decay_power\":2,\"radius\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-endpoint-independent","seed":202611125075,"parameters":{"center":12,"radius":5,"decay_power":0,"domain":"Exact_power_series_radius","operation":"radius-endpoint-independent","semantic_template":"qf-mod43-radius-endpoint-independent/{\"center\":12,\"decay_power\":0,\"radius\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0076","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 1, a_n=1*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125076,"parameters":{"mode":"zero_radius","scale":1,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":1,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0077","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 2, a_n=2/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125077,"parameters":{"mode":"infinite_radius","scale":2,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":2,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0078","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125078,"parameters":{"mode":"finite_prefix","scale":3,"prefix_length":1,"first_ratio":[1,2],"second_ratio":[1,3],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":1,\"scale\":3,\"second_ratio\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0079","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 4, a_n=4*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125079,"parameters":{"mode":"zero_radius","scale":4,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":4,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0080","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 5, a_n=5/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125080,"parameters":{"mode":"infinite_radius","scale":5,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":5,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0081","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125081,"parameters":{"mode":"finite_prefix","scale":6,"prefix_length":2,"first_ratio":[1,2],"second_ratio":[1,4],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":2,\"scale\":6,\"second_ratio\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0082","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 7, a_n=7*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125082,"parameters":{"mode":"zero_radius","scale":7,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":7,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0083","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 8, a_n=8/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125083,"parameters":{"mode":"infinite_radius","scale":8,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":8,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0084","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125084,"parameters":{"mode":"finite_prefix","scale":9,"prefix_length":3,"first_ratio":[1,2],"second_ratio":[1,5],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":3,\"scale\":9,\"second_ratio\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0085","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 10, a_n=10*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125085,"parameters":{"mode":"zero_radius","scale":10,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":10,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0086","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 11, a_n=11/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125086,"parameters":{"mode":"infinite_radius","scale":11,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":11,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125087,"parameters":{"mode":"finite_prefix","scale":12,"prefix_length":4,"first_ratio":[1,2],"second_ratio":[1,6],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":4,\"scale\":12,\"second_ratio\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0088","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 13, a_n=13*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125088,"parameters":{"mode":"zero_radius","scale":13,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":13,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0089","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 14, a_n=14/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125089,"parameters":{"mode":"infinite_radius","scale":14,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":14,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0090","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125090,"parameters":{"mode":"finite_prefix","scale":15,"prefix_length":5,"first_ratio":[1,2],"second_ratio":[1,7],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":5,\"scale\":15,\"second_ratio\":[1,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0091","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 16, a_n=16*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125091,"parameters":{"mode":"zero_radius","scale":16,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":16,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0092","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 17, a_n=17/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125092,"parameters":{"mode":"infinite_radius","scale":17,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":17,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0093","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125093,"parameters":{"mode":"finite_prefix","scale":18,"prefix_length":6,"first_ratio":[1,2],"second_ratio":[1,8],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":6,\"scale\":18,\"second_ratio\":[1,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0094","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 19, a_n=19*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125094,"parameters":{"mode":"zero_radius","scale":19,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":19,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0095","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 20, a_n=20/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125095,"parameters":{"mode":"infinite_radius","scale":20,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":20,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0096","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125096,"parameters":{"mode":"finite_prefix","scale":21,"prefix_length":7,"first_ratio":[1,2],"second_ratio":[1,9],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":7,\"scale\":21,\"second_ratio\":[1,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0097","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 22, a_n=22*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125097,"parameters":{"mode":"zero_radius","scale":22,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":22,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0098","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(infinite_radius, 23, a_n=23/n!_for_every_n>=0, infinity, 是, factorial_denominator_gives_infinite_radius)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125098,"parameters":{"mode":"infinite_radius","scale":23,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"infinite_radius\",\"prefix_length\":0,\"scale\":23,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0099","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125099,"parameters":{"mode":"finite_prefix","scale":24,"prefix_length":8,"first_ratio":[1,2],"second_ratio":[1,10],"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":[1,2],\"mode\":\"finite_prefix\",\"prefix_length\":8,\"scale\":24,\"second_ratio\":[1,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0100","batch_id":"batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations","language":"zh-Hant","source_mko_ids":["mko-power-series-radius-of-convergence"],"title_zh":"冪級數收斂半徑","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_radius, 25, a_n=25*n!_for_every_n>=0, 0, 是, factorial_growth_gives_radius_zero)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["冪級數收斂半徑"]},"variant":{"family_id":"qf-mod43-radius-zero-infinity-prefix-boundary","seed":202611125100,"parameters":{"mode":"zero_radius","scale":25,"prefix_length":0,"first_ratio":1,"second_ratio":1,"domain":"Exact_power_series_radius","operation":"radius-zero-infinity-prefix-boundary","semantic_template":"qf-mod43-radius-zero-infinity-prefix-boundary/{\"first_ratio\":1,\"mode\":\"zero_radius\",\"prefix_length\":0,\"scale\":25,\"second_ratio\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}