{
  "aliases": [
    "madhava-leibniz-series"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0T32ME0A6H"
    }
  ],
  "body": "\n## Statement\n\nThe series $\\sum_{n=0}^{\\infty} \\frac{(-1)^n}{2n + 1}$ converges, and\n$$1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots = \\frac{\\pi}{4}.$$\n\n## Remarks\n\nThe series converges by the [[alternating series test|0PK8QH1W1V]]. Its value comes from the [[power series of arctangent|0T32ME0A6H]] at $x = 1$, since $\\arctan 1 = \\pi/4$.\nThe convergence is very slow. The error after the terms up to $n = N$ is at most $1/(2N + 3)$.\nMathlib states it as a limit of partial sums, in `Real.tendsto_sum_pi_div_four`.\n\n## History\n\nMadhava of Sangamagrama or his school found the series in the 14th or 15th century. James Gregory found it again in 1671, and Leibniz in 1673.\n\n## Explanations\n\n### L1 human\n\nAdd and subtract the reciprocals of the odd numbers: $1 - 1/3 + 1/5 - \\cdots$.\nThe result is a quarter of $\\pi$, which links the odd numbers with circles.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "madhava-leibniz-series"
    ],
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "Leibniz formula for pi",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q97226587",
      "wikipedia": "Leibniz formula for π"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Leibniz_formula_for_π"
      }
    ],
    "tag": "037FD5NBRT",
    "uses": [
      "0PK8QH1W1V",
      "0C1VYVNV2N"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/037FD5NBRT/",
  "links": [
    {
      "field": "uses",
      "target": "0PK8QH1W1V"
    },
    {
      "field": "uses",
      "target": "0C1VYVNV2N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/037FD5NBRT.md",
  "msc": [
    "40A05"
  ],
  "name": "Leibniz formula for pi",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/037/037FD5NBRT.md",
  "status": null,
  "tag": "037FD5NBRT",
  "uri": "m2://037FD5NBRT",
  "url": "https://math2.tamnd.com/wiki/leibniz-formula-for-pi/",
  "writing": {
    "tag": "037FD5NBRT",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}