{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-power-series-radius-of-convergence-01", "mko_id": "mko-power-series-radius-of-convergence", "title": { "zh-Hant": "半徑、端點與完整係數律", "en": "Radius, endpoints and complete coefficient laws" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-power-series-radius-of-convergence-01", "source": { "mko_id": "mko-power-series-radius-of-convergence", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "6c1fd0862cf0aff9deb2796ea76f2af6aa6acd82d07eb53b87a10aab859cde1f", "english_sha256": "f732982ca9bde1233312a021b49651f496d8732ee6037036ac799752c9694e68" }, "mode": "worked_example", "audience_prerequisites": [ "algebra", "epsilon", "geom", "harmonic_boundaries" ], "goal": { "zh-Hant": "對幾何係數 2^n 求 R，並展示端點及 0/infinity 邊界。", "en": "Find R for geometric coefficients 2^n, and exhibit endpoint and zero/infinity boundaries." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "主級數 sum_{n≥0}2^n(x-1)^n；另考慮 endpoint law 2^n/(n+1)。", "en": "The main series is sum over n≥0 of 2^n*(x-1)^n; also consider endpoint coefficients 2^n/(n+1)." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "2^n(x-1)^n", "after_tex": "r^n,\\quad r=2(x-1)", "relation": "equivalent", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "整個冪級數化成幾何族，中心固定 1。", "en": "The full power series becomes geometric with center fixed at 1." }, "reverse_audit": { "kind": "inverse_equivalence", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "係數 2^n 與位移冪一起合併，未丟中心。", "en": "Combine the coefficient and shifted power without losing the center." } }, "leaf_basis": "algebra" }, { "id": "S02", "uses": [ "S01" ], "before_tex": "|x-1|<1/2", "after_tex": "|r|<1\\Longrightarrow\\sum_{n\\ge0}|r|^n<\\infty", "relation": "implies", "rule": "geom", "conditions": [], "explanation": { "zh-Hant": "有限和公式及消失尾項給內部絕對收斂。", "en": "The finite-sum formula and vanishing remainder prove absolute convergence inside." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S01" ], "explanation": { "zh-Hant": "必須用絕對值判內部，不只正 r。", "en": "The interior uses absolute values, not positive r alone." } }, "leaf_basis": "geom" }, { "id": "S03", "uses": [ "S01" ], "before_tex": "|x-1|>1/2", "after_tex": "|r|>1\\Longrightarrow r^n\\not\\to0", "relation": "implies", "rule": "geom", "conditions": [], "explanation": { "zh-Hant": "外部項不趨零，因此級數發散；R=1/2。", "en": "Outside, terms fail to vanish, so the series diverges; thus R=1/2." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S01" ], "explanation": { "zh-Hant": "發散由必要項條件，非僅公式分母符號。", "en": "Divergence uses the necessary term condition, not the sign of a formula's denominator." } }, "leaf_basis": "geom" }, { "id": "S04", "uses": [ "S01", "S02", "S03" ], "before_tex": "x=1+1/2\\text{ or }1-1/2", "after_tex": "r=1\\text{ or }-1;\\quad r^n\\not\\to0", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "主幾何族兩個端點均發散，但這是另核項條件，不由半徑單獨決定。", "en": "Both endpoints of the main geometric family diverge, separately checked by the term condition rather than radius alone." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S01", "S02", "S03" ], "explanation": { "zh-Hant": "未把此族端點狀態外推全部同半徑冪級數。", "en": "These endpoint statuses are not extrapolated to every series with the same radius." } }, "leaf_basis": "algebra" }, { "id": "S05", "uses": [ "G1" ], "before_tex": "a_n=2^n/(n+1),\\quad x=1\\pm1/2", "after_tex": "\\sum1/(n+1)\\text{ diverges};\\quad\\sum(-1)^n/(n+1)\\text{ converges conditionally}", "relation": "implies", "rule": "harmonic_boundaries", "conditions": [], "explanation": { "zh-Hant": "同半徑 endpoint law 在內部受幾何絕對尾界，外部比率最終>1；端點卻一發散一條件收斂。", "en": "For this same-radius law the interior is geometrically dominated and exterior ratios eventually exceed one; endpoints instead give divergence on one side and conditional convergence on the other." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "端點使用獨立調和與交錯證書。", "en": "The endpoints use separate harmonic and alternating certificates." } }, "leaf_basis": "harmonic_boundaries" }, { "id": "S06", "uses": [ "G1" ], "before_tex": "a_n=n!,\\quad x\\ne0\\text{ about center }0", "after_tex": "|(a_{n+1}x^{n+1})/(a_nx^n)|=(n+1)|x|\\ge2\\text{ eventually};\\quad R=0", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "任意非零 x 的項最終倍增，不能趨零；中心 x=0 只有首項。", "en": "For any nonzero x, the magnitudes of successive terms eventually grow by at least a factor of 2 and cannot tend to zero; at the center only the initial term remains." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "R=0 是合法邊界，不需普通有界集合 sup 規則排除。", "en": "R=0 is a legitimate boundary, not excluded by a bounded-real supremum convention." } }, "leaf_basis": "algebra" }, { "id": "S07", "uses": [ "G1" ], "before_tex": "a_n=1/n!,\\quad x\\in\\mathbb R\\text{ fixed}", "after_tex": "|x|/(n+1)\\le1/2\\text{ eventually};\\quad R=\\infty", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "取整數 N>2|x|，尾項受 ratio 1/2 的幾何尾控制，對每個 x 絕對收斂。", "en": "Choose an integer N>2|x|; the tail is dominated by a ratio-1/2 geometric tail, giving absolute convergence at every x." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "x 固定後 N 可依 x，這足夠證逐點全實數收斂半徑無限。", "en": "After fixing x, N may depend on x; this suffices for an infinite convergence radius." } }, "leaf_basis": "algebra" }, { "id": "S08", "uses": [ "G1" ], "before_tex": "a_n=b_n=0\\ (0\\le n\\le N);\\quad a_n=2^{n-N-1},\\ b_n=3^{n-N-1}\\ (n>N)", "after_tex": "R_a=1/2,\\quad R_b=1/3\\text{ despite the same finite prefix}", "relation": "implies", "rule": "geom", "conditions": [], "explanation": { "zh-Hant": "兩條完整律在有限前綴完全相同，之後尾端幾何率不同；有限零因子不改非中心收斂邊界。", "en": "Two full laws share the entire finite prefix but have different geometric tail rates; a finite power factor does not alter convergence away from the center." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "半徑差異來自明示無限尾，而不是推測未知後項。", "en": "The different radii come from declared infinite tails, not guessed future coefficients." } }, "leaf_basis": "geom" } ], "result": { "zh-Hant": "主族 R=1/2 且兩端發散；同半徑可有不同端點，R=0 與 infinity 均合法，有限前綴不決定 R。", "en": "The main radius is 1/2 with both endpoints divergent; endpoint behavior varies at the same radius, zero/infinity are valid, and finite prefixes do not determine R." }, "negative_controls": [ { "claim": { "zh-Hant": "|x-c|=R 必絕對收斂。", "en": "|x-c|=R guarantees absolute convergence." }, "expected": "reject", "reason": { "zh-Hant": "主幾何族端點項不趨零。", "en": "The main geometric endpoints have nonvanishing terms." } }, { "claim": { "zh-Hant": "ratio limit 不存在就沒有收斂半徑。", "en": "A missing ratio limit means no convergence radius exists." }, "expected": "reject", "reason": { "zh-Hant": "a_n=1+(-1)^n/2 的相鄰比交替為 1/3 與 3，極限不存在；但 1/2≤a_n≤3/2 使 |x|<1 受幾何支配，|x|>1 項不趨零，所以半徑 1。", "en": "For a_n=1+(-1)^n/2, adjacent ratios alternate between 1/3 and 3 and have no limit. Yet 1/2≤a_n≤3/2 gives geometric domination for |x|<1 and nonvanishing terms for |x|>1, so the radius is one." } }, { "claim": { "zh-Hant": "前 N 個係數唯一決定半徑。", "en": "The first N coefficients uniquely determine radius." }, "expected": "reject", "reason": { "zh-Hant": "S08 給相同前綴、不同半徑的完整律。", "en": "S08 supplies full laws with the same prefix and different radii." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "對sum_{n≥0} a_n(x-c)^n，R可為0、有限正數或infinity。若|x-c|R則發散；|x-c|=R時不能由R單獨判定。本輪geometric law固定a_n=q^n；endpoint law固定a_n=R^(-n)/(n+1)^p；prefix boundary的兩序列在0≤n≤N皆為0，之後分別為q_1^(n-N-1)與q_2^(n-N-1)。Ratio test只在相應極限存在時可用，不能當一般定義。有限coefficient prefix不足以決定一般radius，且不得以只支援bounded-real supremum的規則排除R=0或infinity。", "en": "For sum over n≥0 of a_n*(x−c)^n, R may be0, finite positive or infinity. Distance less than R gives absolute convergence and distance greater than R gives divergence; equality cannot be decided from R alone. This slice declares geometric coefficients q^n, endpoint coefficients R^(−n)/(n+1)^p, and two complete laws sharing zeros at every index 0≤n≤N before distinct scaled geometric tails. A ratio test needs its relevant limit and is not a general definition. A finite coefficient prefix cannot determine a general radius, and a bounded-real-only supremum rule must not exclude0 or infinity." }, "source": { "mko_id": "mko-power-series-radius-of-convergence", "sha256": "6c1fd0862cf0aff9deb2796ea76f2af6aa6acd82d07eb53b87a10aab859cde1f", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } }, { "id": "algebra", "kind": "prerequisite", "statement": { "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。", "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities." } }, { "id": "epsilon", "kind": "prerequisite", "statement": { "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。", "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals." } }, { "id": "geom", "kind": "prerequisite", "statement": { "zh-Hant": "有限幾何和為 (1-r^(N+1))/(1-r)（r≠1）；|r|<1 時餘項趨零，|r|≥1 的幾何項不趨零。", "en": "The finite geometric sum is (1-r^(N+1))/(1-r) for r≠1; for |r|<1 its remainder vanishes, while for |r|≥1 its terms do not tend to zero." } }, { "id": "harmonic_boundaries", "kind": "prerequisite", "statement": { "zh-Hant": "調和級數以二進位分塊每塊≥1/2 發散；交錯調和以偶奇部分和夾擠收斂。", "en": "Harmonic dyadic blocks each contribute at least 1/2 and diverge; alternating-harmonic parity partial sums bracket a common limit." } } ], "revision_log": [], "numeric_checks": [] }, "presentation": { "md_scope": { "zh-Hant": "完整係數公式才給半徑證書；有限前綴不足。", "en": "A radius certificate uses a full coefficient law; a finite prefix is insufficient." }, "md_symbols": [ [ "R", "允許 0 或 infinity 的半徑", "radius allowing zero or infinity" ], [ "r", "完整幾何比", "full geometric ratio" ] ], "md_error_controls": [ { "title": "端點另算 / Endpoints separately", "zh": "內外半徑判準不處理等號，端點獨立證明。", "en": "Interior/exterior rules do not settle equality; prove endpoints independently." }, { "title": "全律 / Whole law", "zh": "係數規則要涵蓋全部 n。", "en": "The coefficient law must cover every n." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "b1b6d270c26519919c71c61c0d056e80a89da34835bc561d38bb1032610e97fc", "original_source_sha256": "b7e224751e7201b56434d747ff1921e9b689113df97dd8d49367655f50ab91d8", "owner_adopted": true, "bilingual_sync": "complete_source_bound", "formal_Lean": "NOT_RUN", "reader_review": "142_of_142_source_bound", "wording_review": "original_Lead_decisions_resolved", "glossary_reader_paraphrases": [ "mko_definition" ], "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW", "issue_ids": [ "BR142-S184", "BR142-P059" ], "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a", "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite.", "english_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c" } } }