{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0001","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-12+(0)/(n+1)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-12","explanation_zh":"極限L=-12。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081001,"parameters":{"L":-12,"c":0,"d":1,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-12\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0002","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-11+(-2)/(n+2)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-11","explanation_zh":"極限L=-11。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081002,"parameters":{"L":-11,"c":-2,"d":2,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-11\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0003","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-10+(3)/(n+3)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-10","explanation_zh":"極限L=-10。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081003,"parameters":{"L":-10,"c":3,"d":3,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-10\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0004","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-9+(-4)/(n+4)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-9","explanation_zh":"極限L=-9。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081004,"parameters":{"L":-9,"c":-4,"d":4,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-9\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0005","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-8+(5)/(n+5)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-8","explanation_zh":"極限L=-8。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081005,"parameters":{"L":-8,"c":5,"d":5,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-8\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0006","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-7+(-6)/(n+1)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-7","explanation_zh":"極限L=-7。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081006,"parameters":{"L":-7,"c":-6,"d":1,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-7\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0007","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-6+(7)/(n+2)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-6","explanation_zh":"極限L=-6。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081007,"parameters":{"L":-6,"c":7,"d":2,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-6\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0008","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-5+(-8)/(n+3)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-5","explanation_zh":"極限L=-5。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081008,"parameters":{"L":-5,"c":-8,"d":3,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-5\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0009","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-4+(9)/(n+4)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-4","explanation_zh":"極限L=-4。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081009,"parameters":{"L":-4,"c":9,"d":4,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-4\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0010","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-3+(-10)/(n+5)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-3","explanation_zh":"極限L=-3。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081010,"parameters":{"L":-3,"c":-10,"d":5,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-3\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0011","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-2+(11)/(n+1)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-2","explanation_zh":"極限L=-2。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081011,"parameters":{"L":-2,"c":11,"d":1,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-2\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0012","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=-1+(-12)/(n+2)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"-1","explanation_zh":"極限L=-1。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081012,"parameters":{"L":-1,"c":-12,"d":2,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"-1\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0013","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=0+(13)/(n+3)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"0","explanation_zh":"極限L=0。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081013,"parameters":{"L":0,"c":13,"d":3,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"0\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0014","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=1+(-14)/(n+4)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"1","explanation_zh":"極限L=1。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081014,"parameters":{"L":1,"c":-14,"d":4,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"1\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0015","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=2+(15)/(n+5)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"2","explanation_zh":"極限L=2。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081015,"parameters":{"L":2,"c":15,"d":5,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"2\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0016","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=3+(-16)/(n+1)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"3","explanation_zh":"極限L=3。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081016,"parameters":{"L":3,"c":-16,"d":1,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"3\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0017","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=4+(17)/(n+2)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"4","explanation_zh":"極限L=4。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081017,"parameters":{"L":4,"c":17,"d":2,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"4\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0018","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=5+(-18)/(n+3)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"5","explanation_zh":"極限L=5。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081018,"parameters":{"L":5,"c":-18,"d":3,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"5\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0019","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=6+(19)/(n+4)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"6","explanation_zh":"極限L=6。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081019,"parameters":{"L":6,"c":19,"d":4,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"6\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0020","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=7+(-20)/(n+5)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"7","explanation_zh":"極限L=7。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081020,"parameters":{"L":7,"c":-20,"d":5,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"7\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0021","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=8+(21)/(n+1)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"8","explanation_zh":"極限L=8。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081021,"parameters":{"L":8,"c":21,"d":1,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"8\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0022","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=9+(-22)/(n+2)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"9","explanation_zh":"極限L=9。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081022,"parameters":{"L":9,"c":-22,"d":2,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"9\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0023","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=10+(23)/(n+3)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"10","explanation_zh":"極限L=10。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081023,"parameters":{"L":10,"c":23,"d":3,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"10\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0024","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=11+(-24)/(n+4)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"11","explanation_zh":"極限L=11。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081024,"parameters":{"L":11,"c":-24,"d":4,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"11\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0025","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"明示倒數尾的數列極限","stem_zh":"從0起算實數列 a_n=12+(25)/(n+5)，n∈N0，d≥1。求數列極限。","answer":{"kind":"short_text","value":"12","explanation_zh":"極限L=12。此模型的倒數尾趨於零，極限為指定 L；不是以有限項抽樣建立任意數列的普遍證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["明示倒數尾的數列極限"]},"variant":{"family_id":"qf-sequence-limit-reciprocal-value","seed":202610081025,"parameters":{"L":12,"c":25,"d":5,"operation":"reciprocal_sequence_limit","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-reciprocal-value/{\"inputs\":[\"12\"],\"operation\":\"reciprocal_sequence_limit\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0026","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(0)/(n+1)，n∈N0，d≥1。ε=1/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"0","explanation_zh":"最小尾界N=0。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081026,"parameters":{"c":0,"d":1,"eps_n":1,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"constant_tail\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0027","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-2)/(n+2)，n∈N0，d≥1。ε=2/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"1","explanation_zh":"最小尾界N=1。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081027,"parameters":{"c":-2,"d":2,"eps_n":2,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"2\",\"2\",\"1/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0028","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(3)/(n+3)，n∈N0，d≥1。ε=3/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"1","explanation_zh":"最小尾界N=1。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081028,"parameters":{"c":3,"d":3,"eps_n":3,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"3\",\"3\",\"1/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0029","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-4)/(n+4)，n∈N0，d≥1。ε=4/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"0","explanation_zh":"最小尾界N=0。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081029,"parameters":{"c":-4,"d":4,"eps_n":4,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"4\",\"4\",\"4/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0030","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(5)/(n+1)，n∈N0，d≥1。ε=5/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"2","explanation_zh":"最小尾界N=2。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081030,"parameters":{"c":5,"d":1,"eps_n":5,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"5\",\"1\",\"5/2\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0031","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-6)/(n+2)，n∈N0，d≥1。ε=1/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"17","explanation_zh":"最小尾界N=17。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081031,"parameters":{"c":-6,"d":2,"eps_n":1,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"6\",\"2\",\"1/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0032","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(7)/(n+3)，n∈N0，d≥1。ε=2/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"1","explanation_zh":"最小尾界N=1。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081032,"parameters":{"c":7,"d":3,"eps_n":2,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"7\",\"3\",\"2/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0033","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-8)/(n+4)，n∈N0，d≥1。ε=3/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"2","explanation_zh":"最小尾界N=2。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081033,"parameters":{"c":-8,"d":4,"eps_n":3,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"8\",\"4\",\"3/2\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0034","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(9)/(n+1)，n∈N0，d≥1。ε=4/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"6","explanation_zh":"最小尾界N=6。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081034,"parameters":{"c":9,"d":1,"eps_n":4,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"9\",\"1\",\"4/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0035","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-10)/(n+2)，n∈N0，d≥1。ε=5/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"1","explanation_zh":"最小尾界N=1。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081035,"parameters":{"c":-10,"d":2,"eps_n":5,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"10\",\"2\",\"5/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0036","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(11)/(n+3)，n∈N0，d≥1。ε=1/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"20","explanation_zh":"最小尾界N=20。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081036,"parameters":{"c":11,"d":3,"eps_n":1,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"11\",\"3\",\"1/2\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0037","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-12)/(n+4)，n∈N0，d≥1。ε=2/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"15","explanation_zh":"最小尾界N=15。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081037,"parameters":{"c":-12,"d":4,"eps_n":2,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"12\",\"4\",\"2/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0038","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(13)/(n+1)，n∈N0，d≥1。ε=3/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"4","explanation_zh":"最小尾界N=4。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081038,"parameters":{"c":13,"d":1,"eps_n":3,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"13\",\"1\",\"3/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0039","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-14)/(n+2)，n∈N0，d≥1。ε=4/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"6","explanation_zh":"最小尾界N=6。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081039,"parameters":{"c":-14,"d":2,"eps_n":4,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"14\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0040","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(15)/(n+3)，n∈N0，d≥1。ε=5/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"7","explanation_zh":"最小尾界N=7。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081040,"parameters":{"c":15,"d":3,"eps_n":5,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"15\",\"3\",\"5/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0041","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-16)/(n+4)，n∈N0，d≥1。ε=1/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"13","explanation_zh":"最小尾界N=13。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081041,"parameters":{"c":-16,"d":4,"eps_n":1,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"16\",\"4\",\"1/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0042","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(17)/(n+1)，n∈N0，d≥1。ε=2/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"17","explanation_zh":"最小尾界N=17。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081042,"parameters":{"c":17,"d":1,"eps_n":2,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"17\",\"1\",\"1/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0043","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-18)/(n+2)，n∈N0，d≥1。ε=3/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"17","explanation_zh":"最小尾界N=17。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081043,"parameters":{"c":-18,"d":2,"eps_n":3,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"18\",\"2\",\"1/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0044","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(19)/(n+3)，n∈N0，d≥1。ε=4/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"2","explanation_zh":"最小尾界N=2。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081044,"parameters":{"c":19,"d":3,"eps_n":4,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"19\",\"3\",\"4/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0045","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-20)/(n+4)，n∈N0，d≥1。ε=5/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"5","explanation_zh":"最小尾界N=5。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081045,"parameters":{"c":-20,"d":4,"eps_n":5,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"20\",\"4\",\"5/2\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0046","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(21)/(n+1)，n∈N0，d≥1。ε=1/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"63","explanation_zh":"最小尾界N=63。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081046,"parameters":{"c":21,"d":1,"eps_n":1,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"21\",\"1\",\"1/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0047","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-22)/(n+2)，n∈N0，d≥1。ε=2/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"10","explanation_zh":"最小尾界N=10。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081047,"parameters":{"c":-22,"d":2,"eps_n":2,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"22\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0048","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(23)/(n+3)，n∈N0，d≥1。ε=3/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"13","explanation_zh":"最小尾界N=13。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081048,"parameters":{"c":23,"d":3,"eps_n":3,"eps_d":2,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"23\",\"3\",\"3/2\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0049","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(-24)/(n+4)，n∈N0，d≥1。ε=4/3>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"15","explanation_zh":"最小尾界N=15。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081049,"parameters":{"c":-24,"d":4,"eps_n":4,"eps_d":3,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"24\",\"4\",\"4/3\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0050","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"含截止項的最小自然尾界","stem_zh":"從0起算實數列 a_n=L+(25)/(n+1)，n∈N0，d≥1。ε=5/1>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。","answer":{"kind":"short_text","value":"5","explanation_zh":"最小尾界N=5。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["含截止項的最小自然尾界"]},"variant":{"family_id":"qf-sequence-limit-least-cutoff","seed":202610081050,"parameters":{"c":25,"d":1,"eps_n":5,"eps_d":1,"operation":"least_cutoff","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-least-cutoff/{\"inputs\":[\"25\",\"1\",\"5/1\"],\"operation\":\"least_cutoff\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0051","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(0)/(n+1)，n∈N0，d≥1。ε=1/1>0，候選 N=0 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(0, 1)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081051,"parameters":{"c":0,"d":1,"eps_n":1,"eps_d":1,"N":0,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"constant_tail\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0052","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-2)/(n+2)，n∈N0，d≥1。ε=2/2>0，候選 N=0 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(4, 4)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081052,"parameters":{"c":-2,"d":2,"eps_n":2,"eps_d":2,"N":0,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"2\",\"2\",\"1/1\",\"0\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0053","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(3)/(n+3)，n∈N0，d≥1。ε=3/3>0，候選 N=3 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(9, 18)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081053,"parameters":{"c":3,"d":3,"eps_n":3,"eps_d":3,"N":3,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"3\",\"3\",\"1/1\",\"3\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0054","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-4)/(n+4)，n∈N0，d≥1。ε=4/1>0，候選 N=0 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(4, 16)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081054,"parameters":{"c":-4,"d":4,"eps_n":4,"eps_d":1,"N":0,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"4\",\"4\",\"4/1\",\"0\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0055","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(5)/(n+1)，n∈N0，d≥1。ε=5/2>0，候選 N=1 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(10, 10)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081055,"parameters":{"c":5,"d":1,"eps_n":5,"eps_d":2,"N":1,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"5\",\"1\",\"5/2\",\"1\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0056","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-6)/(n+2)，n∈N0，d≥1。ε=1/3>0，候選 N=19 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(18, 21)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081056,"parameters":{"c":-6,"d":2,"eps_n":1,"eps_d":3,"N":19,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"6\",\"2\",\"1/3\",\"19\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0057","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(7)/(n+3)，n∈N0，d≥1。ε=2/1>0，候選 N=1 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(7, 8)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081057,"parameters":{"c":7,"d":3,"eps_n":2,"eps_d":1,"N":1,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"7\",\"3\",\"2/1\",\"1\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0058","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-8)/(n+4)，n∈N0，d≥1。ε=3/2>0，候選 N=1 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(16, 15)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081058,"parameters":{"c":-8,"d":4,"eps_n":3,"eps_d":2,"N":1,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"8\",\"4\",\"3/2\",\"1\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0059","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(9)/(n+1)，n∈N0，d≥1。ε=4/3>0，候選 N=8 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(27, 36)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081059,"parameters":{"c":9,"d":1,"eps_n":4,"eps_d":3,"N":8,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"9\",\"1\",\"4/3\",\"8\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0060","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-10)/(n+2)，n∈N0，d≥1。ε=5/1>0，候選 N=1 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(10, 15)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081060,"parameters":{"c":-10,"d":2,"eps_n":5,"eps_d":1,"N":1,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"10\",\"2\",\"5/1\",\"1\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0061","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(11)/(n+3)，n∈N0，d≥1。ε=1/2>0，候選 N=19 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(22, 22)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081061,"parameters":{"c":11,"d":3,"eps_n":1,"eps_d":2,"N":19,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"11\",\"3\",\"1/2\",\"19\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0062","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-12)/(n+4)，n∈N0，d≥1。ε=2/3>0，候選 N=17 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(36, 42)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081062,"parameters":{"c":-12,"d":4,"eps_n":2,"eps_d":3,"N":17,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"12\",\"4\",\"2/3\",\"17\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0063","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(13)/(n+1)，n∈N0，d≥1。ε=3/1>0，候選 N=4 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(13, 15)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081063,"parameters":{"c":13,"d":1,"eps_n":3,"eps_d":1,"N":4,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"13\",\"1\",\"3/1\",\"4\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0064","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-14)/(n+2)，n∈N0，d≥1。ε=4/2>0，候選 N=5 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(28, 28)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081064,"parameters":{"c":-14,"d":2,"eps_n":4,"eps_d":2,"N":5,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"14\",\"2\",\"2/1\",\"5\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0065","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(15)/(n+3)，n∈N0，d≥1。ε=5/3>0，候選 N=9 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(45, 60)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081065,"parameters":{"c":15,"d":3,"eps_n":5,"eps_d":3,"N":9,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"15\",\"3\",\"5/3\",\"9\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0066","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-16)/(n+4)，n∈N0，d≥1。ε=1/1>0，候選 N=13 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(16, 17)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081066,"parameters":{"c":-16,"d":4,"eps_n":1,"eps_d":1,"N":13,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"16\",\"4\",\"1/1\",\"13\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0067","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(17)/(n+1)，n∈N0，d≥1。ε=2/2>0，候選 N=16 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(34, 34)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081067,"parameters":{"c":17,"d":1,"eps_n":2,"eps_d":2,"N":16,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"17\",\"1\",\"1/1\",\"16\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0068","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-18)/(n+2)，n∈N0，d≥1。ε=3/3>0，候選 N=19 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(54, 63)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081068,"parameters":{"c":-18,"d":2,"eps_n":3,"eps_d":3,"N":19,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"18\",\"2\",\"1/1\",\"19\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0069","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(19)/(n+3)，n∈N0，d≥1。ε=4/1>0，候選 N=2 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(19, 20)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081069,"parameters":{"c":19,"d":3,"eps_n":4,"eps_d":1,"N":2,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"19\",\"3\",\"4/1\",\"2\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0070","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-20)/(n+4)，n∈N0，d≥1。ε=5/2>0，候選 N=4 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(40, 40)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081070,"parameters":{"c":-20,"d":4,"eps_n":5,"eps_d":2,"N":4,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"20\",\"4\",\"5/2\",\"4\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0071","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(21)/(n+1)，n∈N0，d≥1。ε=1/3>0，候選 N=65 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(63, 66)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081071,"parameters":{"c":21,"d":1,"eps_n":1,"eps_d":3,"N":65,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"21\",\"1\",\"1/3\",\"65\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0072","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-22)/(n+2)，n∈N0，d≥1。ε=2/1>0，候選 N=10 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(22, 24)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081072,"parameters":{"c":-22,"d":2,"eps_n":2,"eps_d":1,"N":10,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"22\",\"2\",\"2/1\",\"10\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0073","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(23)/(n+3)，n∈N0，d≥1。ε=3/2>0，候選 N=12 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"否","explanation_zh":"尾界交叉乘積=(46, 45)；嚴格小於=否。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081073,"parameters":{"c":23,"d":3,"eps_n":3,"eps_d":2,"N":12,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"23\",\"3\",\"3/2\",\"12\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0074","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(-24)/(n+4)，n∈N0，d≥1。ε=4/3>0，候選 N=17 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(72, 84)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081074,"parameters":{"c":-24,"d":4,"eps_n":4,"eps_d":3,"N":17,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"24\",\"4\",\"4/3\",\"17\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0075","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"候選自然尾界的嚴格資格","stem_zh":"從0起算實數列 a_n=L+(25)/(n+1)，n∈N0，d≥1。ε=5/1>0，候選 N=5 是否保證每個 n≥N 均有 |a_n−L|<ε？","answer":{"kind":"short_text","value":"是","explanation_zh":"尾界交叉乘積=(25, 30)；嚴格小於=是。比較截止項的單調誤差界，嚴格小於才合格；數值相等不算 ε 內部，也不是任意數列抽樣證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["候選自然尾界的嚴格資格"]},"variant":{"family_id":"qf-sequence-limit-cutoff-validity","seed":202610081075,"parameters":{"c":25,"d":1,"eps_n":5,"eps_d":1,"N":5,"operation":"cutoff_validity","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-cutoff-validity/{\"inputs\":[\"25\",\"1\",\"5/1\",\"5\"],\"operation\":\"cutoff_validity\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0076","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(0)/(n+1)，n∈N0，d≥1。求 n=0 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"0/1","explanation_zh":"項誤差=0/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081076,"parameters":{"c":0,"d":1,"n":0,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"0\",\"1\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0077","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-2)/(n+2)，n∈N0，d≥1。求 n=1 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"2/3","explanation_zh":"項誤差=2/3。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081077,"parameters":{"c":-2,"d":2,"n":1,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"2\",\"3\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0078","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(3)/(n+3)，n∈N0，d≥1。求 n=2 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"3/5","explanation_zh":"項誤差=3/5。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081078,"parameters":{"c":3,"d":3,"n":2,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"3\",\"5\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0079","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-4)/(n+4)，n∈N0，d≥1。求 n=3 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"4/7","explanation_zh":"項誤差=4/7。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081079,"parameters":{"c":-4,"d":4,"n":3,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"4\",\"7\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0080","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(5)/(n+1)，n∈N0，d≥1。求 n=4 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"1/1","explanation_zh":"項誤差=1/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081080,"parameters":{"c":5,"d":1,"n":4,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"5\",\"5\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0081","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-6)/(n+2)，n∈N0，d≥1。求 n=5 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"6/7","explanation_zh":"項誤差=6/7。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081081,"parameters":{"c":-6,"d":2,"n":5,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"6\",\"7\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0082","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(7)/(n+3)，n∈N0，d≥1。求 n=6 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"7/9","explanation_zh":"項誤差=7/9。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081082,"parameters":{"c":7,"d":3,"n":6,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"7\",\"9\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0083","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-8)/(n+4)，n∈N0，d≥1。求 n=0 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"2/1","explanation_zh":"項誤差=2/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081083,"parameters":{"c":-8,"d":4,"n":0,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"8\",\"4\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0084","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(9)/(n+1)，n∈N0，d≥1。求 n=1 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"9/2","explanation_zh":"項誤差=9/2。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081084,"parameters":{"c":9,"d":1,"n":1,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"9\",\"2\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0085","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-10)/(n+2)，n∈N0，d≥1。求 n=2 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"5/2","explanation_zh":"項誤差=5/2。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081085,"parameters":{"c":-10,"d":2,"n":2,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"10\",\"4\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0086","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(11)/(n+3)，n∈N0，d≥1。求 n=3 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"11/6","explanation_zh":"項誤差=11/6。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081086,"parameters":{"c":11,"d":3,"n":3,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"11\",\"6\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0087","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-12)/(n+4)，n∈N0，d≥1。求 n=4 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"3/2","explanation_zh":"項誤差=3/2。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081087,"parameters":{"c":-12,"d":4,"n":4,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"12\",\"8\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0088","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(13)/(n+1)，n∈N0，d≥1。求 n=5 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"13/6","explanation_zh":"項誤差=13/6。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081088,"parameters":{"c":13,"d":1,"n":5,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"13\",\"6\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0089","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-14)/(n+2)，n∈N0，d≥1。求 n=6 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"7/4","explanation_zh":"項誤差=7/4。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081089,"parameters":{"c":-14,"d":2,"n":6,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"14\",\"8\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0090","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(15)/(n+3)，n∈N0，d≥1。求 n=0 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"5/1","explanation_zh":"項誤差=5/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081090,"parameters":{"c":15,"d":3,"n":0,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"15\",\"3\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0091","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-16)/(n+4)，n∈N0，d≥1。求 n=1 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"16/5","explanation_zh":"項誤差=16/5。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081091,"parameters":{"c":-16,"d":4,"n":1,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"16\",\"5\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0092","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(17)/(n+1)，n∈N0，d≥1。求 n=2 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"17/3","explanation_zh":"項誤差=17/3。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081092,"parameters":{"c":17,"d":1,"n":2,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"17\",\"3\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0093","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-18)/(n+2)，n∈N0，d≥1。求 n=3 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"18/5","explanation_zh":"項誤差=18/5。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081093,"parameters":{"c":-18,"d":2,"n":3,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"18\",\"5\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0094","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(19)/(n+3)，n∈N0，d≥1。求 n=4 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"19/7","explanation_zh":"項誤差=19/7。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081094,"parameters":{"c":19,"d":3,"n":4,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"19\",\"7\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0095","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-20)/(n+4)，n∈N0，d≥1。求 n=5 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"20/9","explanation_zh":"項誤差=20/9。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081095,"parameters":{"c":-20,"d":4,"n":5,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"20\",\"9\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0096","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(21)/(n+1)，n∈N0，d≥1。求 n=6 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"3/1","explanation_zh":"項誤差=3/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081096,"parameters":{"c":21,"d":1,"n":6,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"21\",\"7\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0097","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-22)/(n+2)，n∈N0，d≥1。求 n=0 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"11/1","explanation_zh":"項誤差=11/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081097,"parameters":{"c":-22,"d":2,"n":0,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"22\",\"2\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0098","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(23)/(n+3)，n∈N0，d≥1。求 n=1 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"23/4","explanation_zh":"項誤差=23/4。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081098,"parameters":{"c":23,"d":3,"n":1,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"23\",\"4\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0099","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(-24)/(n+4)，n∈N0，d≥1。求 n=2 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"4/1","explanation_zh":"項誤差=4/1。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081099,"parameters":{"c":-24,"d":4,"n":2,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"24\",\"6\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r01-sequence-limit-0100","batch_id":"batch-20261007-advance-20261008-r01-sequence-limit-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence-limit"],"title_zh":"指定索引的精確項誤差","stem_zh":"從0起算實數列 a_n=L+(25)/(n+1)，n∈N0，d≥1。求 n=3 時 |a_n−L|，採最簡正分母 p/q。","answer":{"kind":"expression","value":"25/4","explanation_zh":"項誤差=25/4。取 c 的絕對值再除以正的 n+d；L 不影響這個誤差。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["指定索引的精確項誤差"]},"variant":{"family_id":"qf-sequence-limit-term-error","seed":202610081100,"parameters":{"c":25,"d":1,"n":3,"operation":"sequence_term_error","domain":"naturals_from_zero","semantic_template":"qf-sequence-limit-term-error/{\"inputs\":[\"25\",\"4\"],\"operation\":\"sequence_term_error\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}