{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0091", "pilot_number": "P1000-0766", "number": 766, "family_id": "qf-mod48-ratio-test-boundary", "title": { "zh-Hant": "第 766 題：級數比值判別法", "en": "Question 766: Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r02-ratio-test-series-0091", "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-ratio-test-series" ], "title_zh": "級數比值判別法", "stem_zh": "eventually nonzero、absolute successive ratio與limit存在責任。操作=ratio-test-boundary；資料={\"mode\":\"signed_without_absolute\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, 否)", "explanation_zh": "Wave6第十一層核對=(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, 否)；依完整明示資料獨立重算；L<1絕對收斂、L>1發散、L=1無結論；有限ratio samples、zero denominators或raw signed ratio不能冒充假設。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "級數比值判別法" ] }, "variant": { "family_id": "qf-mod48-ratio-test-boundary", "seed": 202611172091, "parameters": { "mode": "signed_without_absolute", "scale": 17, "domain": "Exact_ratio_test_series", "operation": "ratio-test-boundary", "semantic_template": "ratio-test-boundary|{mode:\"signed_without_absolute\",scale:17}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0091", "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "65b93df547ce14097c5aaa54a5e6d8d719b5b0e9ebf8be4c845ce35ec70187cb", "family_id": "qf-mod48-ratio-test-boundary", "seed": 202611172091, "semantic_template": "ratio-test-boundary|{mode:\"signed_without_absolute\",scale:17}" }, "fields": { "title": "Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises", "stem": "Use mode=signed_without_absolute and positive integer scale=17. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim).", "answer": { "kind": "expression", "value": "(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)", "explanation": "Retain the specified sequence or ratio pattern. Infinitely many zero denominators, a finite favorable ratio prefix, a nonexistent limit or using a raw signed ratio omit required premises. In the signed example the absolute ratio is1/2 and the series can converge; the final No rejects that raw-ratio method, not the true series status. The exact result, retaining the requested order, is (signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, No)." }, "skills": [ "Zero Denominators, Finite Samples and Signed Ratios Do Not Supply the Ratio-Test Premises" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "eventually nonzero、absolute successive ratio與limit存在責任。操作=ratio-test-boundary；資料={\"mode\":\"signed_without_absolute\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=signed_without_absolute and positive integer scale=17. Give (mode, scale, series_or_ratio_description, scope_or_obstruction, method_application_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "boundary mode=signed_without_absolute; scale=17", "after": "check the precise hypothesis that fails", "relation": "implies", "rule_id": "ratio_hypothesis_audit", "conditions": [], "explanation": { "zh-Hant": "先說明失敗的是哪個假設，不把所有失敗情形都判成發散。", "en": "Identify the failing hypothesis instead of declaring every boundary case divergent." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。", "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "a_n=(-1)^n·17·(1/2)^n", "after": "a_(n+1)/a_n=−1/2", "relation": "implies", "rule_id": "signed_ratio", "conditions": [], "explanation": { "zh-Hant": "先正確算有號比值，負號不能在判別條件中忽略。", "en": "Compute the signed ratio correctly; its negative sign cannot be ignored in the hypotheses." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。", "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "|a_(n+1)/a_n|", "after": "absolute ratio=1/2<1", "relation": "implies", "rule_id": "absolute_ratio_required", "conditions": [], "explanation": { "zh-Hant": "取絕對值後可套正確的絕對收斂判別；本題false是否定錯用，不是否定此級數收斂。", "en": "The absolute ratio licenses the correct convergence test; the false flag rejects misuse, not convergence of this series." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。", "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "missing or misused premise", "after": "this attempted test is not licensed", "relation": "implies", "rule_id": "boundary_verdict", "conditions": [], "explanation": { "zh-Hant": "保留原題邊界標記，不偷加未證的一般收斂結論。", "en": "Retain the original boundary verdict without adding an unsupported general convergence conclusion." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "有限核對不證無限尾端；本步的量詞、非零分母與絕對值條件仍須保留。", "en": "Finite checks do not prove an infinite tail; retain this step's quantifiers, nonzero denominators and absolute values." } }, "arithmetic": null } ], "math_model": { "result": [ "signed_without_absolute", 17, "a_n=(-1)^n*17*(1/2)^n", "raw_ratio=-1/2_but_absolute_ratio=1/2", false ] }, "conclusion": { "canonical_value": "(signed_without_absolute, 17, a_n=(-1)^n*17*(1/2)^n, raw_ratio=-1/2_but_absolute_ratio=1/2, 否)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "比值判別需要最終非零尾端、絕對值比值與真正存在的極限；邊界題指出失敗的假設，不判定級數本身。", "en": "The ratio test needs an eventually nonzero tail, absolute ratios and an existing limit. Boundary items name the failed hypothesis without deciding the series." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations", "batch_path": "public/data/questions/revisions/ocme-notation-guard-revision-20261018/batch-20261015-advance-20261117-r02-ratio-test-series-foundations/questions.jsonl", "batch_sha256": "bb49d7293d2866c4dbd0d9b5ea035c809d3b1a31966971f6cb354eea49d79996", "original_line": 91, "original_line_sha256": "cb1a954fc439fec69247c64c4066b37ae7ac0539b450720bc9aeb51aebe60098", "question_compact_sha256": "65b93df547ce14097c5aaa54a5e6d8d719b5b0e9ebf8be4c845ce35ec70187cb", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r02-ratio-test-series-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r02-ratio-test-series-foundations.en.jsonl", "english_row_compact_sha256": "ae3da8836ed880b0bc918bad7edee21b2a41d3b67573bb5b71c131019f13652b" }, "mko_bindings": { "mko-ratio-test-series": { "sha256": "90d6b9fa7fcadbd04ee6ef7efff16b13647e3de4bf46c8c2248ca8168ee49382", "english_sha256": "4cedc70d3305cd8238c8d23bc1dca04541d313107ee78217fb5a79d376d7ff3e", "original_path": "public/data/mko/ratio-test-series.json", "english_path": "public/data/i18n/editions/mko-ratio-test-series.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "cece8f797d92c4c086a96974ae38f06602bbbd69c54beb24032b9a6a5c66b7e8", "original_candidate_line_sha256": "cece8f797d92c4c086a96974ae38f06602bbbd69c54beb24032b9a6a5c66b7e8" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-003-03", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c7b8b7b89507961341d86c7ea2d54dc766db76ac3ad20a1aafac5e703e64dc7e", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }