{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261103-r03-infinite-series-0083", "pilot_number": "R0034-17783", "number": 17783, "family_id": "qf-mod34-series-term-zero-boundary", "title": { "zh-Hant": "第 17783 題：無窮級數", "en": "Question 17783: Terms Tending to Zero Do Not Make a Harmonic Tail Summable" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261103-r03-infinite-series-0083", "batch_id": "batch-20261014-advance-20261103-r03-infinite-series-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3a2ae6971a39917fe4ed3c14abb68bb51ff9d97ab02b2a6a38fd237e645a896b", "family_id": "qf-mod34-series-term-zero-boundary", "seed": 202611033083, "semantic_template": "qf-mod34-series-term-zero-boundary/{\"prefix_length\":5,\"shift\":8}" }, "fields": { "title": "Terms Tending to Zero Do Not Make a Harmonic Tail Summable", "stem": "Use the shifted harmonic terms 1/(n+shift), n=0,1,..., with positive shift=8 and displayed prefix_length=5. Give (shift,prefix_length,terms,terms_tend_zero?,series_converges?,terms_tend_to_zero_but_shifted_harmonic_series_diverges).", "answer": { "kind": "expression", "value": "(8, 5, [1/8, 1/9, 1/10, 1/11, 1/12], Yes, No, terms_tend_to_zero_but_shifted_harmonic_series_diverges)", "explanation": "The displayed terms are only a finite prefix, while the named shifted-harmonic argument supplies the divergence conclusion. Terms tend to zero, a necessary but not sufficient condition for series convergence. Do not confuse the limit of terms with the limit of partial sums. The exact result, retaining the requested order, is (8, 5, [1/8, 1/9, 1/10, 1/11, 1/12], Yes, No, terms_tend_to_zero_but_shifted_harmonic_series_diverges)." }, "skills": [ "Terms Tending to Zero Do Not Make a Harmonic Tail Summable" ], "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": "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。", "en": "Use the shifted harmonic terms 1/(n+shift), n=0,1,..., with positive shift=8 and displayed prefix_length=5. Give (shift,prefix_length,terms,terms_tend_zero?,series_converges?,terms_tend_to_zero_but_shifted_harmonic_series_diverges)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "harmonic displayed index0 denominator := sum(0, 8)", "after": "harmonic displayed index0 denominator = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index0 denominator，不以小數近似替代。", "en": "Recompute harmonic displayed index0 denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "8" ], "output": "8", "label": "harmonic displayed index0 denominator" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "harmonic displayed index0 term := div(1, 8)", "after": "harmonic displayed index0 term = 1/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index0 term，不以小數近似替代。", "en": "Recompute harmonic displayed index0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "8" ], "output": "1/8", "label": "harmonic displayed index0 term" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "harmonic displayed index1 denominator := sum(1, 8)", "after": "harmonic displayed index1 denominator = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index1 denominator，不以小數近似替代。", "en": "Recompute harmonic displayed index1 denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "8" ], "output": "9", "label": "harmonic displayed index1 denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "harmonic displayed index1 term := div(1, 9)", "after": "harmonic displayed index1 term = 1/9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index1 term，不以小數近似替代。", "en": "Recompute harmonic displayed index1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "9" ], "output": "1/9", "label": "harmonic displayed index1 term" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "harmonic displayed index2 denominator := sum(2, 8)", "after": "harmonic displayed index2 denominator = 10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index2 denominator，不以小數近似替代。", "en": "Recompute harmonic displayed index2 denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "8" ], "output": "10", "label": "harmonic displayed index2 denominator" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "harmonic displayed index2 term := div(1, 10)", "after": "harmonic displayed index2 term = 1/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index2 term，不以小數近似替代。", "en": "Recompute harmonic displayed index2 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "10" ], "output": "1/10", "label": "harmonic displayed index2 term" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "harmonic displayed index3 denominator := sum(3, 8)", "after": "harmonic displayed index3 denominator = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index3 denominator，不以小數近似替代。", "en": "Recompute harmonic displayed index3 denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "8" ], "output": "11", "label": "harmonic displayed index3 denominator" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "harmonic displayed index3 term := div(1, 11)", "after": "harmonic displayed index3 term = 1/11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index3 term，不以小數近似替代。", "en": "Recompute harmonic displayed index3 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "11" ], "output": "1/11", "label": "harmonic displayed index3 term" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "harmonic displayed index4 denominator := sum(4, 8)", "after": "harmonic displayed index4 denominator = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index4 denominator，不以小數近似替代。", "en": "Recompute harmonic displayed index4 denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "8" ], "output": "12", "label": "harmonic displayed index4 denominator" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "harmonic displayed index4 term := div(1, 12)", "after": "harmonic displayed index4 term = 1/12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 harmonic displayed index4 term，不以小數近似替代。", "en": "Recompute harmonic displayed index4 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "12" ], "output": "1/12", "label": "harmonic displayed index4 term" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "shifted harmonic series:DECLARED whole positive tail b_k=1/(k+8),k>=0;shift>0", "after": "b_k decreases to0:for ANY epsilon>0 choose N with N+shift>1/epsilon,then every k>=N has b_k=0 large enough that2^j>=shift-1,the block m=2^j+1..2^(j+1) has2^j terms each>=1/2^(j+1);each block sum>=1/2;infinitely many disjoint blocks make partial sums unbounded", "relation": "implies", "rule_id": "harmonic_shifted_dyadic_blocks_diverge_not_term_limit", "conditions": [], "explanation": { "zh-Hant": "整個尾和只少有限開頭；每個充分晚的dyadic區塊至少1/2，任取多個不交塊總和無界，才證級數發散。項趨零沒有給partial sums的有限極限。", "en": "Removing finitely many initial terms leaves the dyadic-block argument.Each sufficiently late block contributes at least1/2,so arbitrarily many disjoint blocks force unbounded partial sums.Terms tending to zero do not give a finite limit of partial sums." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "displayed finite prefix=[1/8, 1/9, 1/10, 1/11, 1/12];SEPARATE complete harmonic-tail proof retained", "after": "terms tend to0=是;series converges=否;terms_tend_to_zero_but_shifted_harmonic_series_diverges", "relation": "implies", "rule_id": "series_term_limit_not_partial_sum_limit", "conditions": [], "explanation": { "zh-Hant": "有限清單只供顯示，兩個無限命題由明示全尾與區塊論證分別給出；必要的項趨零不充分。", "en": "The finite list is illustrative.The two infinite claims come separately from the declared full tail and block argument;the necessary term-zero condition is not sufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 8, 5, [ "1/8", "1/9", "1/10", "1/11", "1/12" ], true, false, "terms_tend_to_zero_but_shifted_harmonic_series_diverges" ] }, "conclusion": { "canonical_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-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261103-r03-infinite-series-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r03-infinite-series-foundations/questions.jsonl", "batch_sha256": "312c02ec0cfac82c5f3d25ced63c9d3a9b9b58bd4a3fae40456906f9b2dab03c", "original_line": 83, "original_line_sha256": "1f7165ae6b6e419e6edb6a5ef80965edf979a8a7282d4a40388bbea4fdc46499", "question_compact_sha256": "3a2ae6971a39917fe4ed3c14abb68bb51ff9d97ab02b2a6a38fd237e645a896b", "english_row_compact_sha256": "64b1ce4ecc55cc6cda0ca4d87a172579b7bf62127db2868e4bc37f72cb8c8236", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r03-infinite-series-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r03-infinite-series-foundations.en.jsonl" }, "mko_bindings": { "mko-infinite-series": { "sha256": "f2c97043e047957807c3f303d9bf37bbe3d188f1721ae0dcfbab1537a45b8d0b", "english_sha256": "d540bafe796e5c209c969a5f34c55ef21c4244a2805d9558a5cbc8938ef09592", "original_path": "public/data/mko/infinite-series.json", "english_path": "public/data/i18n/editions/mko-infinite-series.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2", "original_candidate_sha256": "85c7ffacd6015222bc829f982f5e272cbab0223c5fa55dc0538e836346ca6bf2", "original_candidate_line_sha256": "85c7ffacd6015222bc829f982f5e272cbab0223c5fa55dc0538e836346ca6bf2" }, "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": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-008-p35", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336", "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 13, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "3bbdefc87c5d542e5ccea4f93b617fd0e324dc97cbc883d9dd513086d3015770", "arithmetic_steps": 10 }, "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." } }