{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0KC0F51T6A"
    }
  ],
  "body": "\n## Statement\n\nThe partial products of the following infinite product converge to $\\pi/2$:\n$$\\frac{\\pi}{2} = \\prod_{n=1}^{\\infty} \\frac{4 n^2}{4 n^2 - 1} = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdots.$$\n\n## Remarks\n\nLet $I_n = \\int_0^{\\pi/2} \\sin^n x \\, dx$. The [[reduction formula|012RYET5E2]] gives $I_{2n}$ and $I_{2n+1}$ as explicit products.\nThe sequence $I_n$ is decreasing, and $I_{2n+1} / I_{2n} \\to 1$ by the [[squeeze theorem|0XMF00T4XQ]]. This ratio is a partial product times $2/\\pi$.\nMathlib states it as `Real.tendsto_prod_pi_div_two`.\n\n## History\n\nJohn Wallis published the product in 1656.\n\n## Explanations\n\n### L1 human\n\nThe Wallis product writes $\\pi/2$ as an infinite product of simple fractions.\nThe partial products converge slowly to $\\pi/2$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "11Y60"
    ],
    "name": "Wallis product",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1501324",
      "wikipedia": "Wallis product"
    },
    "sources": [
      {
        "locator": "Proof using integration",
        "url": "https://en.wikipedia.org/wiki/Wallis_product"
      }
    ],
    "tag": "0XCYA4YERS",
    "uses": [
      "012RYET5E2",
      "0XMF00T4XQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0XCYA4YERS/",
  "links": [
    {
      "field": "uses",
      "target": "012RYET5E2"
    },
    {
      "field": "uses",
      "target": "0XMF00T4XQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0XCYA4YERS.md",
  "msc": [
    "11Y60"
  ],
  "name": "Wallis product",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0XC/0XCYA4YERS.md",
  "status": null,
  "tag": "0XCYA4YERS",
  "uri": "m2://0XCYA4YERS",
  "url": "https://math2.tamnd.com/wiki/wallis-product/",
  "writing": {
    "tag": "0XCYA4YERS",
    "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
      }
    ]
  }
}