{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0085", "pilot_number": "R0034-17985", "number": 17985, "family_id": "qf-mod35-absseries-term-zero-boundary", "title": { "zh-Hant": "第 17985 題：絕對收斂", "en": "Question 17985: Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261104-r03-absolute-convergence-of-series-0085", "batch_id": "batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-absolute-convergence-of-series" ], "title_zh": "絕對收斂", "stem_zh": "partial sums與absolute-series exact convergence certificate。操作=absseries-term-zero-boundary；資料={\"shift\":10,\"prefix_length\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(10, 5, 是, 否, term_zero_not_absolute_convergence)", "explanation_zh": "Wave5第四層核對=(10, 5, 是, 否, term_zero_not_absolute_convergence)；依完整明示資料重算；absolute convergence核sum absolute values並推出原級數收斂；term趨零 bounded或alternation不足；conditional需雙證書。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "絕對收斂" ] }, "variant": { "family_id": "qf-mod35-absseries-term-zero-boundary", "seed": 202611043085, "parameters": { "shift": 10, "prefix_length": 5, "domain": "Exact_absolute_or_conditional_series", "operation": "absseries-term-zero-boundary", "semantic_template": "qf-mod35-absseries-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": "候選須主端重算；不構成人工審定、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或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-20261104-r03-absolute-convergence-of-series-0085", "batch_id": "batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6c5055b3ab5d5fb4e03462c3ab1d954c3e5e276afe76931869303944d17bce95", "family_id": "qf-mod35-absseries-term-zero-boundary", "seed": 202611043085, "semantic_template": "qf-mod35-absseries-term-zero-boundary/{\"prefix_length\":5,\"shift\":10}" }, "fields": { "title": "Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion", "stem": "Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary).", "answer": { "kind": "expression", "value": "(10, 5, Yes, No, term_zero_not_absolute_convergence)", "explanation": "The harmonic terms tend to zero, but that condition alone does not supply absolute convergence. The No is a sufficiency boundary, not a proof obtained merely by inspecting the finite prefix; the harmonic comparison supplies the counterexample. The exact result, retaining the requested order, is (10, 5, Yes, No, term_zero_not_absolute_convergence)." }, "skills": [ "Terms Tending to Zero Are Not a Sufficient Absolute-Convergence Criterion" ], "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": "partial sums與absolute-series exact convergence certificate。操作=absseries-term-zero-boundary；資料={\"shift\":10,\"prefix_length\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the shifted-harmonic boundary with positive shift=10 and inspected prefix_length=5. Give (shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=absseries-term-zero-boundary;original question snapshot retained;source presentation interface resolved in this page", "after": "EXPLICIT ordered output fields=[shift, prefix_length, terms_tend_to_zero, term_zero_suffices_for_absolute_convergence, boundary]", "relation": "implies", "rule_id": "absseries_explicit_source_interface_fields_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記，這裡直接列完整欄位，不需瀏覽器JavaScript或外部查表。欄位名只是presentation metadata，不是數學運算；term-zero的第四個否是『條件是否充分』，不是誤標成另一個converges旗標。", "en": "Preserve the original snapshot's historical interface marker while listing the full ordered fields here,without browser JavaScript or an external lookup.Field labels are presentation metadata,not mathematical evaluation.The fourth term-zero No marks insufficiency,not a mislabeled second convergence flag." }, "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": "S002", "uses": [ "G001", "S001" ], "before": "absolute-convergence term-zero counterexample:DECLARED whole positive tail b_k=1/(k+10),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": "S004", "uses": [ "G001", "S003" ], "before": "inspected finite prefix length=5;DECLARED harmonic tail shift=10;whole-tail term limit0 and divergent absolute sum proved separately", "after": "terms_tend_to_zero=是;term_zero_suffices_for_absolute_convergence=否;term_zero_not_absolute_convergence", "relation": "implies", "rule_id": "absseries_term_zero_insufficiency_marker_not_prefix_classification", "conditions": [], "explanation": { "zh-Hant": "第四個否回答項趨零是否充分，不是從這段prefix做一次絕對收斂測試；完整調和反例已給項趨零但絕對值級數不收斂。", "en": "The fourth No answers whether the term-zero condition suffices,not a convergence test inferred from this prefix.The complete harmonic counterexample has terms tending to0 while its absolute series diverges." }, "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": [ 10, 5, true, false, "term_zero_not_absolute_convergence" ] }, "conclusion": { "canonical_value": "(10, 5, 是, 否, term_zero_not_absolute_convergence)", "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-20261104-r03-absolute-convergence-of-series-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations/questions.jsonl", "batch_sha256": "80786ddf5c8f2dd129df5ad6a139679d6f762f4ad7b6c20caa9920af15a19506", "original_line": 85, "original_line_sha256": "e9ff6997b38557e9405488859e689224f48e92afd97915f36e527ec0f6cc2344", "question_compact_sha256": "6c5055b3ab5d5fb4e03462c3ab1d954c3e5e276afe76931869303944d17bce95", "english_row_compact_sha256": "acc87e4e5c4237c4c516dad68fbd5b62392110b184065f2a829a08b73fd20bd9", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r03-absolute-convergence-of-series-foundations.en.jsonl" }, "mko_bindings": { "mko-absolute-convergence-of-series": { "sha256": "4419bd0abbb51b07728caad18022eaace6aec3c8243880972704e8b1413cc1f8", "english_sha256": "faeb3d8b3f3ed278c06df7c7a387e9df82810c62c55a1d90becc7eaa1efaee1c", "original_path": "public/data/mko/absolute-convergence-of-series.json", "english_path": "public/data/i18n/editions/mko-absolute-convergence-of-series.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2", "original_candidate_sha256": "4d3aad41650d734bb1b717dbef19daae718132ef316a8f3cd0b1034485756a17", "original_candidate_line_sha256": "4d3aad41650d734bb1b717dbef19daae718132ef316a8f3cd0b1034485756a17" }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bdb98d500a07f1a5a06871c7cc5f2f0304dc31f1fa5f492eeef2aae504ba6acd", "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." } }