{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r02-ratio-test-series-0054", "pilot_number": "P1000-0729", "number": 729, "family_id": "qf-mod48-ratio-limit-one-inconclusive", "title": { "zh-Hant": "第 729 題：級數比值判別法", "en": "Question 729: Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r02-ratio-test-series-0054", "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-limit-one-inconclusive；資料={\"exponent\":2,\"shift\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)", "explanation_zh": "Wave6第十一層核對=(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)；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": 1, "skills": [ "級數比值判別法" ] }, "variant": { "family_id": "qf-mod48-ratio-limit-one-inconclusive", "seed": 202611172054, "parameters": { "exponent": 2, "shift": 4, "domain": "Exact_ratio_test_series", "operation": "ratio-limit-one-inconclusive", "semantic_template": "ratio-limit-one-inconclusive|{exponent:2,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-0054", "batch_id": "batch-20261015-advance-20261117-r02-ratio-test-series-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "84e7ff9e6a48e3d5d58c680eec55722d0e21d419a8d670f4774d1ca2bd181326", "family_id": "qf-mod48-ratio-limit-one-inconclusive", "seed": 202611172054, "semantic_template": "ratio-limit-one-inconclusive|{exponent:2,shift:4}" }, "fields": { "title": "Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence", "stem": "Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=4≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope).", "answer": { "kind": "expression", "value": "(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)", "explanation": "The nonconstant absolute ratio is ((n+shift)/(n+shift+1))^p and tends to1 in both cases. The known harmonic-tail/p-series statuses differ, so L=1 itself decides neither. Retain those named reference facts rather than pretending finite ratios prove a limit. The exact result, retaining the requested order, is (2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)." }, "skills": [ "Ratio Limit One Allows Both Harmonic Divergence and Squared-Reciprocal Convergence" ], "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-limit-one-inconclusive；資料={\"exponent\":2,\"shift\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use a_n=1/(n+shift)^p with p=2∈{1,2}, shift=4≥1, n≥0. Give (exponent, shift, absolute_ratio_description, ratio_limit, actual_series_status, ratio_test_scope)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "a_n=1/(n+4)^2, n≥0", "after": "|a_(n+1)/a_n|=((n+4)/(n+5))^2", "relation": "implies", "rule_id": "successive_ratio", "conditions": [], "explanation": { "zh-Hant": "先用完整正項公式推導相鄰絕對比值。", "en": "Derive the absolute ratio from the full positive-term formula." }, "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": "(n+4)/(n+5)=1−1/(n+5)", "after": "ratio limit=1", "relation": "implies", "rule_id": "rational_ratio_limit", "conditions": [], "explanation": { "zh-Hant": "分母增大使相對誤差趨零；不是分母趨零。", "en": "The growing denominator makes the relative error vanish; the denominator itself does not vanish." }, "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": "k≥2", "after": "1/k² ≤1/(k(k−1)) =1/(k−1)−1/k", "relation": "implies", "rule_id": "telescoping_majorant", "conditions": [], "explanation": { "zh-Hant": "正分母下k(k−1)≤k²，取倒數後得到上界。", "en": "For positive denominators k(k−1)≤k², and reciprocation gives the stated upper bound." }, "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": "sum k=K through M, K=max(s,2)", "after": "sum 1/k² ≤1/(K−1)−1/M ≤1/(K−1)", "relation": "implies", "rule_id": "bounded_positive_partials", "conditions": [], "explanation": { "zh-Hant": "尾端有完整望遠鏡上界；若s=1另加第一項1。正部分和單調有界而收斂。", "en": "The tail has a telescoping bound; if s=1 add the first term1. Positive partial sums are increasing and bounded, hence converge." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "same ratio limit1 for both exponent families", "after": "L=1 alone does not decide convergence", "relation": "implies", "rule_id": "inconclusive_ratio_boundary", "conditions": [], "explanation": { "zh-Hant": "真正分類由上述獨立的全尾端論證給出，不是由比值極限1給出。", "en": "The classification comes from the separate complete-tail arguments above, not from the ratio limit1." }, "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": [ 2, 4, "abs_ratio=((n+4)/(n+5))^2", 1, "convergent_p_series", "L_equal_one_is_inconclusive" ] }, "conclusion": { "canonical_value": "(2, 4, abs_ratio=((n+4)/(n+5))^2, 1, convergent_p_series, L_equal_one_is_inconclusive)", "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": "p=1 與 p=2 的比值極限都是 1；分類來自調和區塊與望遠鏡上界，比值判別無結論。", "en": "For p=1 and p=2 the ratio limit is 1; the classification comes from harmonic blocks and a telescoping bound, so the ratio test is inconclusive." }, "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": 54, "original_line_sha256": "72e1895819590199fb87249e861b8c76986e332f82fe4260d677f925319e6600", "question_compact_sha256": "84e7ff9e6a48e3d5d58c680eec55722d0e21d419a8d670f4774d1ca2bd181326", "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": "6896b0e3c33ae7b24a75398a63943c1413440590a4178de93579fd1171d69611" }, "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": "50f3e3d4c8c5a1f0ff49ebaec037679baf3fda2ff717ed90f0ab9d993d971193", "original_candidate_line_sha256": "50f3e3d4c8c5a1f0ff49ebaec037679baf3fda2ff717ed90f0ab9d993d971193" }, "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-20261026-003-02", "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": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "10842ac11036301cfdf2b4c34bb5f1adb91bfb67f18c1cb7a9066ba3ea467bbe", "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." } }