{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0001","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,2],[-1,3]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/2, -1/3], 1/6, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/2, -1/3], 1/6, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033001,"parameters":{"terms":[[1,2],[-1,3]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,2],[-1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0002","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,3],[-1,4]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/3, -1/4], 1/12, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/3, -1/4], 1/12, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033002,"parameters":{"terms":[[1,3],[-1,4]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,3],[-1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0003","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,4],[-1,5]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/4, -1/5], 1/20, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/4, -1/5], 1/20, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033003,"parameters":{"terms":[[1,4],[-1,5]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,4],[-1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0004","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,5],[-1,6]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/5, -1/6], 1/30, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/5, -1/6], 1/30, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033004,"parameters":{"terms":[[1,5],[-1,6]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,5],[-1,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0005","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,6],[-1,7]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/6, -1/7], 1/42, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/6, -1/7], 1/42, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033005,"parameters":{"terms":[[1,6],[-1,7]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,6],[-1,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0006","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,7],[-1,8]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/7, -1/8], 1/56, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/7, -1/8], 1/56, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033006,"parameters":{"terms":[[1,7],[-1,8]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,7],[-1,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0007","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,8],[-1,9]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/8, -1/9], 1/72, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/8, -1/9], 1/72, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033007,"parameters":{"terms":[[1,8],[-1,9]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,8],[-1,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0008","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,9],[-1,10]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/9, -1/10], 1/90, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/9, -1/10], 1/90, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033008,"parameters":{"terms":[[1,9],[-1,10]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,9],[-1,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0009","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,10],[-1,11]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/10, -1/11], 1/110, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/10, -1/11], 1/110, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033009,"parameters":{"terms":[[1,10],[-1,11]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,10],[-1,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0010","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,11],[-1,12]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/11, -1/12], 1/132, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/11, -1/12], 1/132, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033010,"parameters":{"terms":[[1,11],[-1,12]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,11],[-1,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0011","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,12],[-1,13]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/12, -1/13], 1/156, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/12, -1/13], 1/156, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033011,"parameters":{"terms":[[1,12],[-1,13]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,12],[-1,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0012","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,13],[-1,14]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/13, -1/14], 1/182, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/13, -1/14], 1/182, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033012,"parameters":{"terms":[[1,13],[-1,14]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,13],[-1,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0013","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,14],[-1,15]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/14, -1/15], 1/210, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/14, -1/15], 1/210, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033013,"parameters":{"terms":[[1,14],[-1,15]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,14],[-1,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0014","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,15],[-1,16]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/15, -1/16], 1/240, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/15, -1/16], 1/240, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033014,"parameters":{"terms":[[1,15],[-1,16]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,15],[-1,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0015","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,16],[-1,17]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/16, -1/17], 1/272, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/16, -1/17], 1/272, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033015,"parameters":{"terms":[[1,16],[-1,17]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,16],[-1,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0016","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,17],[-1,18]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/17, -1/18], 1/306, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/17, -1/18], 1/306, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033016,"parameters":{"terms":[[1,17],[-1,18]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,17],[-1,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0017","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,18],[-1,19]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/18, -1/19], 1/342, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/18, -1/19], 1/342, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033017,"parameters":{"terms":[[1,18],[-1,19]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,18],[-1,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0018","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,19],[-1,20]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/19, -1/20], 1/380, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/19, -1/20], 1/380, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033018,"parameters":{"terms":[[1,19],[-1,20]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,19],[-1,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0019","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,20],[-1,21]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/20, -1/21], 1/420, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/20, -1/21], 1/420, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033019,"parameters":{"terms":[[1,20],[-1,21]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,20],[-1,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0020","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,21],[-1,22]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/21, -1/22], 1/462, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/21, -1/22], 1/462, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033020,"parameters":{"terms":[[1,21],[-1,22]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,21],[-1,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0021","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,22],[-1,23]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/22, -1/23], 1/506, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/22, -1/23], 1/506, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033021,"parameters":{"terms":[[1,22],[-1,23]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,22],[-1,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0022","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,23],[-1,24]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/23, -1/24], 1/552, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/23, -1/24], 1/552, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033022,"parameters":{"terms":[[1,23],[-1,24]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,23],[-1,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0023","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,24],[-1,25]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/24, -1/25], 1/600, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/24, -1/25], 1/600, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033023,"parameters":{"terms":[[1,24],[-1,25]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,24],[-1,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0024","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,25],[-1,26]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/25, -1/26], 1/650, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/25, -1/26], 1/650, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033024,"parameters":{"terms":[[1,25],[-1,26]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,25],[-1,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0025","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-partial-sum；資料={\"terms\":[[1,26],[-1,27]]}。請依 [terms,partial_sum,count,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/26, -1/27], 1/702, 2, finite_partial_sum_only)","explanation_zh":"Wave5第三層核對=([1/26, -1/27], 1/702, 2, finite_partial_sum_only)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-partial-sum","seed":202611033025,"parameters":{"terms":[[1,26],[-1,27]],"domain":"Exact_symbolic_series_certificate","operation":"series-partial-sum","semantic_template":"qf-mod34-series-partial-sum/{\"terms\":[[1,26],[-1,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0026","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[1,2],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1/2, 1/2, 1, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(1/2, 1/2, 1, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033026,"parameters":{"a":[1,2],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[1,2],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0027","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[2,3],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(2/3, 1/2, 4/3, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033027,"parameters":{"a":[2,3],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[2,3],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0028","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[3,4],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3/4, 1/2, 3/2, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(3/4, 1/2, 3/2, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033028,"parameters":{"a":[3,4],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[3,4],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0029","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[4,5],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4/5, 1/2, 8/5, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(4/5, 1/2, 8/5, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033029,"parameters":{"a":[4,5],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[4,5],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0030","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[5,6],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(5/6, 1/2, 5/3, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(5/6, 1/2, 5/3, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033030,"parameters":{"a":[5,6],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[5,6],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0031","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[6,7],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(6/7, 1/2, 12/7, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(6/7, 1/2, 12/7, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033031,"parameters":{"a":[6,7],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[6,7],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0032","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[7,8],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(7/8, 1/2, 7/4, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(7/8, 1/2, 7/4, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033032,"parameters":{"a":[7,8],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[7,8],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0033","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[8,9],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(8/9, 1/2, 16/9, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(8/9, 1/2, 16/9, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033033,"parameters":{"a":[8,9],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[8,9],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0034","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[9,10],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(9/10, 1/2, 9/5, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(9/10, 1/2, 9/5, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033034,"parameters":{"a":[9,10],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[9,10],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0035","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[10,11],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(10/11, 1/2, 20/11, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(10/11, 1/2, 20/11, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033035,"parameters":{"a":[10,11],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[10,11],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0036","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[11,12],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(11/12, 1/2, 11/6, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(11/12, 1/2, 11/6, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033036,"parameters":{"a":[11,12],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[11,12],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0037","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[12,13],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(12/13, 1/2, 24/13, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(12/13, 1/2, 24/13, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033037,"parameters":{"a":[12,13],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[12,13],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0038","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[13,14],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(13/14, 1/2, 13/7, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(13/14, 1/2, 13/7, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033038,"parameters":{"a":[13,14],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[13,14],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0039","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[14,15],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(14/15, 1/2, 28/15, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(14/15, 1/2, 28/15, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033039,"parameters":{"a":[14,15],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[14,15],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0040","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[15,16],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(15/16, 1/2, 15/8, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(15/16, 1/2, 15/8, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033040,"parameters":{"a":[15,16],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[15,16],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0041","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[16,17],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(16/17, 1/2, 32/17, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(16/17, 1/2, 32/17, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033041,"parameters":{"a":[16,17],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[16,17],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0042","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[17,18],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(17/18, 1/2, 17/9, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(17/18, 1/2, 17/9, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033042,"parameters":{"a":[17,18],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[17,18],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0043","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[18,19],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(18/19, 1/2, 36/19, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(18/19, 1/2, 36/19, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033043,"parameters":{"a":[18,19],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[18,19],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0044","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[19,20],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(19/20, 1/2, 19/10, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(19/20, 1/2, 19/10, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033044,"parameters":{"a":[19,20],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[19,20],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0045","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[20,21],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(20/21, 1/2, 40/21, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(20/21, 1/2, 40/21, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033045,"parameters":{"a":[20,21],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[20,21],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0046","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[21,22],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(21/22, 1/2, 21/11, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(21/22, 1/2, 21/11, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033046,"parameters":{"a":[21,22],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[21,22],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0047","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[22,23],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(22/23, 1/2, 44/23, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(22/23, 1/2, 44/23, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033047,"parameters":{"a":[22,23],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[22,23],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0048","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[23,24],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(23/24, 1/2, 23/12, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(23/24, 1/2, 23/12, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033048,"parameters":{"a":[23,24],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[23,24],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0049","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[24,25],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(24/25, 1/2, 48/25, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(24/25, 1/2, 48/25, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033049,"parameters":{"a":[24,25],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[24,25],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0050","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-geometric-sum；資料={\"a\":[25,26],\"r\":[1,2]}。請依 [a,r,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(25/26, 1/2, 25/13, absolute_ratio_below_one_symbolic_tail)","explanation_zh":"Wave5第三層核對=(25/26, 1/2, 25/13, absolute_ratio_below_one_symbolic_tail)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-geometric-sum","seed":202611033050,"parameters":{"a":[25,26],"r":[1,2],"domain":"Exact_symbolic_series_certificate","operation":"series-geometric-sum","semantic_template":"qf-mod34-series-geometric-sum/{\"a\":[25,26],\"r\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0051","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":1,\"shift\":2,\"n\":1}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 1, 1/6, 1/3, 1/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(1, 2, 1, 1/6, 1/3, 1/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033051,"parameters":{"c":1,"shift":2,"n":1,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":1,\"n\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0052","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":2,\"shift\":2,\"n\":2}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 1/2, 2/4, 2/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(2, 2, 2, 1/2, 2/4, 2/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033052,"parameters":{"c":2,"shift":2,"n":2,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":2,\"n\":2,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0053","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":3,\"shift\":2,\"n\":3}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 3, 9/10, 3/5, 3/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(3, 2, 3, 9/10, 3/5, 3/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033053,"parameters":{"c":3,"shift":2,"n":3,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":3,\"n\":3,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0054","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":4,\"shift\":2,\"n\":4}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 4, 4/3, 4/6, 4/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(4, 2, 4, 4/3, 4/6, 4/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033054,"parameters":{"c":4,"shift":2,"n":4,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":4,\"n\":4,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0055","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":5,\"shift\":2,\"n\":5}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 5, 25/14, 5/7, 5/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(5, 2, 5, 25/14, 5/7, 5/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033055,"parameters":{"c":5,"shift":2,"n":5,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":5,\"n\":5,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0056","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":6,\"shift\":2,\"n\":6}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 6, 9/4, 6/8, 6/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(6, 2, 6, 9/4, 6/8, 6/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033056,"parameters":{"c":6,"shift":2,"n":6,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":6,\"n\":6,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0057","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":7,\"shift\":2,\"n\":7}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(7, 2, 7, 49/18, 7/9, 7/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(7, 2, 7, 49/18, 7/9, 7/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033057,"parameters":{"c":7,"shift":2,"n":7,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":7,\"n\":7,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0058","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":8,\"shift\":2,\"n\":8}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 8, 16/5, 8/10, 8/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(8, 2, 8, 16/5, 8/10, 8/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033058,"parameters":{"c":8,"shift":2,"n":8,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":8,\"n\":8,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0059","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":9,\"shift\":2,\"n\":9}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(9, 2, 9, 81/22, 9/11, 9/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(9, 2, 9, 81/22, 9/11, 9/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033059,"parameters":{"c":9,"shift":2,"n":9,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":9,\"n\":9,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0060","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":10,\"shift\":2,\"n\":10}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(10, 2, 10, 25/6, 10/12, 10/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(10, 2, 10, 25/6, 10/12, 10/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033060,"parameters":{"c":10,"shift":2,"n":10,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":10,\"n\":10,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0061","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":11,\"shift\":2,\"n\":11}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(11, 2, 11, 121/26, 11/13, 11/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(11, 2, 11, 121/26, 11/13, 11/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033061,"parameters":{"c":11,"shift":2,"n":11,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":11,\"n\":11,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0062","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":12,\"shift\":2,\"n\":12}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 12, 36/7, 12/14, 12/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(12, 2, 12, 36/7, 12/14, 12/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033062,"parameters":{"c":12,"shift":2,"n":12,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":12,\"n\":12,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0063","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":13,\"shift\":2,\"n\":13}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(13, 2, 13, 169/30, 13/15, 13/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(13, 2, 13, 169/30, 13/15, 13/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033063,"parameters":{"c":13,"shift":2,"n":13,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":13,\"n\":13,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0064","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":14,\"shift\":2,\"n\":14}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(14, 2, 14, 49/8, 14/16, 14/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(14, 2, 14, 49/8, 14/16, 14/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033064,"parameters":{"c":14,"shift":2,"n":14,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":14,\"n\":14,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0065","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":15,\"shift\":2,\"n\":15}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(15, 2, 15, 225/34, 15/17, 15/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(15, 2, 15, 225/34, 15/17, 15/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033065,"parameters":{"c":15,"shift":2,"n":15,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":15,\"n\":15,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0066","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":16,\"shift\":2,\"n\":16}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(16, 2, 16, 64/9, 16/18, 16/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(16, 2, 16, 64/9, 16/18, 16/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033066,"parameters":{"c":16,"shift":2,"n":16,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":16,\"n\":16,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0067","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":17,\"shift\":2,\"n\":17}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(17, 2, 17, 289/38, 17/19, 17/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(17, 2, 17, 289/38, 17/19, 17/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033067,"parameters":{"c":17,"shift":2,"n":17,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":17,\"n\":17,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0068","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":18,\"shift\":2,\"n\":18}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(18, 2, 18, 81/10, 18/20, 18/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(18, 2, 18, 81/10, 18/20, 18/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033068,"parameters":{"c":18,"shift":2,"n":18,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":18,\"n\":18,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0069","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":19,\"shift\":2,\"n\":19}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(19, 2, 19, 361/42, 19/21, 19/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(19, 2, 19, 361/42, 19/21, 19/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033069,"parameters":{"c":19,"shift":2,"n":19,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":19,\"n\":19,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0070","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":20,\"shift\":2,\"n\":20}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(20, 2, 20, 100/11, 20/22, 20/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(20, 2, 20, 100/11, 20/22, 20/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033070,"parameters":{"c":20,"shift":2,"n":20,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":20,\"n\":20,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0071","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":21,\"shift\":2,\"n\":21}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(21, 2, 21, 441/46, 21/23, 21/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(21, 2, 21, 441/46, 21/23, 21/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033071,"parameters":{"c":21,"shift":2,"n":21,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":21,\"n\":21,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0072","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":22,\"shift\":2,\"n\":22}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(22, 2, 22, 121/12, 22/24, 22/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(22, 2, 22, 121/12, 22/24, 22/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033072,"parameters":{"c":22,"shift":2,"n":22,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":22,\"n\":22,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0073","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":23,\"shift\":2,\"n\":23}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(23, 2, 23, 529/50, 23/25, 23/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(23, 2, 23, 529/50, 23/25, 23/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033073,"parameters":{"c":23,"shift":2,"n":23,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":23,\"n\":23,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0074","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":24,\"shift\":2,\"n\":24}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(24, 2, 24, 144/13, 24/26, 24/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(24, 2, 24, 144/13, 24/26, 24/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033074,"parameters":{"c":24,"shift":2,"n":24,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":24,\"n\":24,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0075","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-telescoping-sum；資料={\"c\":25,\"shift\":2,\"n\":25}。請依 [c,shift,n,partial_sum,remainder,infinite_sum,certificate] 欄序回答。","answer":{"kind":"expression","value":"(25, 2, 25, 625/54, 25/27, 25/2, exact_telescoping_remainder)","explanation_zh":"Wave5第三層核對=(25, 2, 25, 625/54, 25/27, 25/2, exact_telescoping_remainder)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-telescoping-sum","seed":202611033075,"parameters":{"c":25,"shift":2,"n":25,"domain":"Exact_symbolic_series_certificate","operation":"series-telescoping-sum","semantic_template":"qf-mod34-series-telescoping-sum/{\"c\":25,\"n\":25,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0076","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":1,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 5, [1, 1/2, 1/3, 1/4, 1/5], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(1, 5, [1, 1/2, 1/3, 1/4, 1/5], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033076,"parameters":{"shift":1,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0077","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":2,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 5, [1/2, 1/3, 1/4, 1/5, 1/6], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(2, 5, [1/2, 1/3, 1/4, 1/5, 1/6], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033077,"parameters":{"shift":2,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0078","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":3,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [1/3, 1/4, 1/5, 1/6, 1/7], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(3, 5, [1/3, 1/4, 1/5, 1/6, 1/7], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033078,"parameters":{"shift":3,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0079","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":4,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [1/4, 1/5, 1/6, 1/7, 1/8], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(4, 5, [1/4, 1/5, 1/6, 1/7, 1/8], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033079,"parameters":{"shift":4,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0080","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":5,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [1/5, 1/6, 1/7, 1/8, 1/9], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(5, 5, [1/5, 1/6, 1/7, 1/8, 1/9], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033080,"parameters":{"shift":5,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0081","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":6,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6, 5, [1/6, 1/7, 1/8, 1/9, 1/10], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(6, 5, [1/6, 1/7, 1/8, 1/9, 1/10], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033081,"parameters":{"shift":6,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0082","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":7,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7, 5, [1/7, 1/8, 1/9, 1/10, 1/11], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(7, 5, [1/7, 1/8, 1/9, 1/10, 1/11], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033082,"parameters":{"shift":7,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0083","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":8,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8, 5, [1/8, 1/9, 1/10, 1/11, 1/12], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(8, 5, [1/8, 1/9, 1/10, 1/11, 1/12], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033083,"parameters":{"shift":8,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0084","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":9,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9, 5, [1/9, 1/10, 1/11, 1/12, 1/13], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(9, 5, [1/9, 1/10, 1/11, 1/12, 1/13], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033084,"parameters":{"shift":9,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0085","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":10,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10, 5, [1/10, 1/11, 1/12, 1/13, 1/14], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(10, 5, [1/10, 1/11, 1/12, 1/13, 1/14], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033085,"parameters":{"shift":10,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0086","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":11,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11, 5, [1/11, 1/12, 1/13, 1/14, 1/15], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(11, 5, [1/11, 1/12, 1/13, 1/14, 1/15], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033086,"parameters":{"shift":11,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0087","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":12,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12, 5, [1/12, 1/13, 1/14, 1/15, 1/16], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(12, 5, [1/12, 1/13, 1/14, 1/15, 1/16], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033087,"parameters":{"shift":12,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0088","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":13,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13, 5, [1/13, 1/14, 1/15, 1/16, 1/17], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(13, 5, [1/13, 1/14, 1/15, 1/16, 1/17], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033088,"parameters":{"shift":13,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0089","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":14,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14, 5, [1/14, 1/15, 1/16, 1/17, 1/18], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(14, 5, [1/14, 1/15, 1/16, 1/17, 1/18], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033089,"parameters":{"shift":14,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0090","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":15,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15, 5, [1/15, 1/16, 1/17, 1/18, 1/19], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(15, 5, [1/15, 1/16, 1/17, 1/18, 1/19], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033090,"parameters":{"shift":15,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0091","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":16,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16, 5, [1/16, 1/17, 1/18, 1/19, 1/20], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(16, 5, [1/16, 1/17, 1/18, 1/19, 1/20], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033091,"parameters":{"shift":16,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0092","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":17,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17, 5, [1/17, 1/18, 1/19, 1/20, 1/21], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(17, 5, [1/17, 1/18, 1/19, 1/20, 1/21], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033092,"parameters":{"shift":17,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0093","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":18,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18, 5, [1/18, 1/19, 1/20, 1/21, 1/22], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(18, 5, [1/18, 1/19, 1/20, 1/21, 1/22], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033093,"parameters":{"shift":18,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0094","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":19,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19, 5, [1/19, 1/20, 1/21, 1/22, 1/23], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(19, 5, [1/19, 1/20, 1/21, 1/22, 1/23], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033094,"parameters":{"shift":19,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0095","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":20,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20, 5, [1/20, 1/21, 1/22, 1/23, 1/24], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(20, 5, [1/20, 1/21, 1/22, 1/23, 1/24], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033095,"parameters":{"shift":20,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0096","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":21,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21, 5, [1/21, 1/22, 1/23, 1/24, 1/25], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(21, 5, [1/21, 1/22, 1/23, 1/24, 1/25], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033096,"parameters":{"shift":21,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0097","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":22,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22, 5, [1/22, 1/23, 1/24, 1/25, 1/26], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(22, 5, [1/22, 1/23, 1/24, 1/25, 1/26], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033097,"parameters":{"shift":22,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0098","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":23,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23, 5, [1/23, 1/24, 1/25, 1/26, 1/27], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(23, 5, [1/23, 1/24, 1/25, 1/26, 1/27], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033098,"parameters":{"shift":23,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0099","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":24,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24, 5, [1/24, 1/25, 1/26, 1/27, 1/28], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(24, 5, [1/24, 1/25, 1/26, 1/27, 1/28], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033099,"parameters":{"shift":24,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r03-infinite-series-0100","batch_id":"batch-20261014-advance-20261103-r03-infinite-series-foundations","language":"zh-Hant","source_mko_ids":["mko-infinite-series"],"title_zh":"無窮級數","stem_zh":"固定index convention的partial-sum sequence與exact tail。操作=series-term-zero-boundary；資料={\"shift\":25,\"prefix_length\":5}。請依 [shift,prefix_length,terms,terms_tend_zero,series_converges,boundary] 欄序回答。","answer":{"kind":"expression","value":"(25, 5, [1/25, 1/26, 1/27, 1/28, 1/29], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)","explanation_zh":"Wave5第三層核對=(25, 5, [1/25, 1/26, 1/27, 1/28, 1/29], 是, 否, terms_tend_to_zero_but_shifted_harmonic_series_diverges)；依完整明示資料重算；geometric要求絕對ratio小於1且telescoping保留remainder；項趨零必要但不充分；長有限prefix不證收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["無窮級數"]},"variant":{"family_id":"qf-mod34-series-term-zero-boundary","seed":202611033100,"parameters":{"shift":25,"prefix_length":5,"domain":"Exact_symbolic_series_certificate","operation":"series-term-zero-boundary","semantic_template":"qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}