{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-ratio-test-series",
  "semantic_family_id": "sf-ratio-test-series",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "級數比值判別法"
  },
  "summary": {
    "zh-Hant": "非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。"
  },
  "statement": {
    "zh-Hant": "對eventually nonzero序列明示L=lim |a_(n+1)/a_n|。L<1時另取L<q<1並建立eventual geometric domination而得absolute convergence；L>1時項絕對值最終不趨零而發散；L=1時判別無結論，p-series的p=2與p=1提供相反結果。必須取absolute ratio、證明極限而非只算有限多項，zero terms與不存在的ratio limit另行處理。"
  },
  "formula": {
    "tex": "R(a,L)=c",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>R</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>L</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "L",
            "glyph": "L"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "c",
        "glyph": "c"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c3121d0d4d9fca1c19ac56ee36e2a3156b7b32d25ee1c309f8252ed9898ce275"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "相鄰比在某個尾端定義且其絕對值極限存在。"
    },
    {
      "id": "a2",
      "zh-Hant": "L與1的比較使用exact rational或明示symbolic limit certificate。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "ratio_test_operator",
      "role_zh": "比值判別算子",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "series_terms",
      "role_zh": "級數項",
      "scope": "representative_formula"
    },
    {
      "token": "L",
      "role": "absolute_ratio_limit",
      "role_zh": "絕對比極限",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "ratio_test_conclusion",
      "role_zh": "判別結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-absolute-convergence-of-series",
      "title_zh": "級數的絕對收斂",
      "reason_zh": "L<1的結論與級數語義。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "ratio limit。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "絕對比。"
    },
    {
      "id": "mko-series-comparison-test",
      "title_zh": "級數比較判別法",
      "reason_zh": "L<q<1後的geometric domination。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核eventual nonzero與absolute ratio limit。",
      "L<1時明示L<q<1與geometric domination；L>1與L=1另行分層。",
      "非恆定p-series ratios提供L=1相反witness；有限ratio samples與limit不存在都不得冒充定理假設。"
    ],
    "common_misconception_zh": "把L=1當收斂、忽略絕對值，或由前幾個ratio就宣稱極限。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "geometric與p-series witnesses不取代一般ratio-test proof。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入root test與limsup版本。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-ratio-test-series.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-17 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}