{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-infinite-series",
  "semantic_family_id": "sf-infinite-series",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "無窮級數"
  },
  "summary": {
    "zh-Hant": "無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。"
  },
  "statement": {
    "zh-Hant": "本輪處理精確幾何級數與望遠鏡級數。保留起始index、partial sum與remainder；幾何級數只在|r|<1使用sum公式。長有限prefix與a_n→0皆不足以單獨證收斂。"
  },
  "formula": {
    "tex": "S(a)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>S</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "S",
        "glyph": "S"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ddef874d4fed39c285201eece7de68063ea68b4718360eb6e418698eba75f9c7"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "項與partial sums使用精確有理數。"
    },
    {
      "id": "a2",
      "zh-Hant": "symbolic tail明示適用的所有n。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "infinite_series",
      "role_zh": "級數紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "term_sequence",
      "role_zh": "項數列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "項與partial sums。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "級數收斂。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "算術運算",
      "reason_zh": "精確求和。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "partial sums而非單項定義級數。",
      "幾何和保留|r|<1。",
      "telescoping保留remainder；a_n→0與finite prefix不足。"
    ],
    "common_misconception_zh": "只因項趨零或前很多項穩定就宣稱級數收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核明示geometric/telescoping symbolic family。",
    "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absolute convergence與power series。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-infinite-series.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-03 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"
  }
}