{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-absolute-convergence-of-series",
  "semantic_family_id": "sf-absolute-convergence-of-series",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "級數絕對收斂"
  },
  "summary": {
    "zh-Hant": "sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。"
  },
  "statement": {
    "zh-Hant": "幾何級數用|r|<1證絕對收斂；交錯harmonic以alternating remainder證原級數收斂、以harmonic divergence證非絕對。bounded terms、term→0、|partial sum|皆不足。"
  },
  "formula": {
    "tex": "A(S)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "S",
          "glyph": "S"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3d67f237f5a6df6c0543715300a149cf1c8163dc34dc3f9d97ad51060c75583d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "固定index與exact rational partial sums/tails。"
    },
    {
      "id": "a2",
      "zh-Hant": "conditional例子同時有convergence與absolute-divergence證書。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "absolute_series_convergence",
      "role_zh": "絕對收斂",
      "scope": "representative_formula"
    },
    {
      "token": "S",
      "role": "series",
      "role_zh": "級數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-infinite-series",
      "title_zh": "無窮級數",
      "reason_zh": "partial sums。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "absolute series。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "兩級數收斂。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "核sum |a_n|而非|sum a_n|。",
      "absolute convergence implies convergence。",
      "alternation alone不夠；conditional需雙證書。"
    ],
    "common_misconception_zh": "由term→0、bounded terms或alternation直接判absolute convergence。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理明示geometric與alternating-harmonic families。",
    "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comparison tests與power series。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-absolute-convergence-of-series.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-04 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}