{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075", "pilot_number": "R0046-23825", "number": 23825, "family_id": "qf-mod43-radius-endpoint-independent", "title": { "zh-Hant": "第 23825 題：冪級數收斂半徑", "en": "Question 23825: The Positive and Negative Endpoints Can Have Different Status under the Same Radius" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0075", "batch_id": "batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-power-series-radius-of-convergence" ], "title_zh": "冪級數收斂半徑", "stem_zh": "明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={\"center\":12,\"radius\":5,\"decay_power\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)", "explanation_zh": "Wave6第六層核對=(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)；依完整明示資料獨立重算；geometric採a_n=q^n；endpoint採a_n=R^(-n)/(n+1)^p；prefix boundary明示共同zero prefix及兩條完整geometric tail laws；inside絕對收斂、outside發散而endpoints逐案判定；R=0與R=infinity均合法，有限prefix不能決定radius。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "冪級數收斂半徑" ] }, "variant": { "family_id": "qf-mod43-radius-endpoint-independent", "seed": 202611125075, "parameters": { "center": 12, "radius": 5, "decay_power": 0, "domain": "Exact_power_series_radius", "operation": "radius-endpoint-independent", "semantic_template": "qf-mod43-radius-endpoint-independent/{\"center\":12,\"decay_power\":0,\"radius\":5}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r05-power-series-radius-of-convergence-0075", "batch_id": "batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "84720cfce7e658c577e32faffb50d70a3d3611f8e6de2250bb421cacc7933eb0", "family_id": "qf-mod43-radius-endpoint-independent", "seed": 202611125075, "semantic_template": "qf-mod43-radius-endpoint-independent/{\"center\":12,\"decay_power\":0,\"radius\":5}" }, "fields": { "title": "The Positive and Negative Endpoints Can Have Different Status under the Same Radius", "stem": "Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=12, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary).", "answer": { "kind": "expression", "value": "(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)", "explanation": "At c+R the terms are1/(n+1)^p; at c−R they alternate. At p=0 neither converges; p=1 gives positive-endpoint divergence and negative-endpoint conditional convergence; p=2 gives absolute convergence at both. The radius alone does not determine these outcomes. The exact result, retaining the requested order, is (a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, Yes, same_radius_does_not_determine_either_endpoint)." }, "skills": [ "The Positive and Negative Endpoints Can Have Different Status under the Same Radius" ], "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": "明示對每個n>=0有效的完整coefficient law、center與extended radius。操作=radius-endpoint-independent；資料={\"center\":12,\"radius\":5,\"decay_power\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use a_n=5^(−n)/(n+1)^0 for every n≥0 and center=12, with radius R=5>0 and decay_power in {0,1,2}. Give (coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=radius-endpoint-independent;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[coefficient_law, center, radius, decay_power, positive_endpoint_status, negative_endpoint_status, separate_endpoint_certificate, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題介面快照，本頁直接列來源欄序，不需JavaScript。radius由原mode選6／11欄，不補欄；No與實際否定依來源責任分清。", "en": "Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations." }, "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": "ORIGINAL center=12;R=5>0;power=0;WHOLE a_n=R^(-n)/(n+1)^power for EVERY n>=0", "after": "coefficient ratio tends1/R,so radius=5;at c+R terms=1/(n+1)^0;at c-R terms=(-1)^n/(n+1)^0", "relation": "implies", "rule_id": "radius_ENDPOINT_whole_coefficient_law_same_radius", "conditions": [], "explanation": { "zh-Hant": "完整law給ratio limit與半徑，但兩端仍要代值；不是從有限prefix猜root-test極限。", "en": "The whole law gives the ratio limit and radius;substitute at each endpoint separately,not a root-test limit guessed from a prefix." }, "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": "S003", "uses": [ "G001", "S002" ], "before": "ORIGINAL endpoint power=0;positive endpoint=c+R,negative endpoint=c-R", "after": "positive_endpoint_status=divergence;negative_endpoint_status=divergence_terms_do_not_tend_to_zero;separate_endpoint_certificate=是", "relation": "implies", "rule_id": "radius_TWO_endpoints_separate_term_pseries_alternating_tests", "conditions": [], "explanation": { "zh-Hant": "p0兩端項皆不趨0；p1正端harmonic發散、負端交錯項遞減至0而收斂，但絕對harmonic仍發散；p2兩端皆由p-series絕對收斂。同一R不決定這些結果。", "en": "For p0,terms fail to tend to0 at either endpoint.For p1,the positive harmonic series diverges while the alternating series converges conditionally.For p2,both converge absolutely by the p-series test.The radius alone determines none of these endpoint outcomes." }, "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": [ "a_n=5^(-n)/(n+1)^0_for_every_n>=0", "12", 5, 0, "divergence", "divergence_terms_do_not_tend_to_zero", true, "same_radius_does_not_determine_either_endpoint" ] }, "conclusion": { "canonical_value": "(a_n=5^(-n)/(n+1)^0_for_every_n>=0, 12, 5, 0, divergence, divergence_terms_do_not_tend_to_zero, 是, same_radius_does_not_determine_either_endpoint)", "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-20261112-r05-power-series-radius-of-convergence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/questions.jsonl", "batch_sha256": "75ea72f67b08f0d64822fea279c968391c8a15e19b1c10230cf82af81dabe3f4", "original_line": 75, "original_line_sha256": "c1f64c6b73b9189aef5564508a25d401dcc7a7b05003d7e67306b388b7173c6b", "question_compact_sha256": "84720cfce7e658c577e32faffb50d70a3d3611f8e6de2250bb421cacc7933eb0", "english_row_compact_sha256": "ddd5fccdc7c4db714b50901a20c35b020558ef9689e8125b00ffce2711718f88", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations.en.jsonl" }, "mko_bindings": { "mko-power-series-radius-of-convergence": { "sha256": "6c1fd0862cf0aff9deb2796ea76f2af6aa6acd82d07eb53b87a10aab859cde1f", "english_sha256": "f732982ca9bde1233312a021b49651f496d8732ee6037036ac799752c9694e68", "original_path": "public/data/mko/power-series-radius-of-convergence.json", "english_path": "public/data/i18n/editions/mko-power-series-radius-of-convergence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "04d5fe00084f382c9e6853d5f11c80b558d9cf857160ceb644fa20b64948d1fe", "original_candidate_sha256": "9e5ffb09392e0da7e10681a0764b4c2e7d9063f66c54c7f6b00689c8c73136cf", "original_candidate_line_sha256": "9e5ffb09392e0da7e10681a0764b4c2e7d9063f66c54c7f6b00689c8c73136cf", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-power-series-radius-of-convergence": "1905cc950ea16efa2bba777857906fac8a5629d35e8fe96a78832a5f09343413" } } }, "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-20261026-009-p47", "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": "a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1", "question_first_witness_sha256": "c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1e77d652ac17fa0b8920cc150fbf07750a1ad65fc36dc6eabeb6c93188cba174", "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." } }