{
  "aliases": [
    "cauchy-hadamard-formula"
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  "body": "\n## Statement\n\nLet $\\sum a_n (x - c)^n$ be a real power series with [[radius of convergence|0660QX1VPR]] $R$.\nThen\n$$\\frac{1}{R} = \\limsup_{n \\to \\infty} |a_n|^{1/n},$$\nwith the conventions $1/0 = \\infty$ and $1/\\infty = 0$.\n\n## Remarks\n\nThe proof applies the [[root test|05H4DDDYBZ]] to the series $\\sum a_n (x - c)^n$. The [[limit superior|0P8BA7T7S5]] of $|a_n (x - c)^n|^{1/n}$ is $|x - c| \\limsup |a_n|^{1/n}$.\nFor example, $\\sum n x^n$ has $R = 1$, since $n^{1/n} \\to 1$.\nMathlib states it as `FormalMultilinearSeries.radius_inv_eq_limsup`.\n\n## Explanations\n\n### L1 human\n\nThe coefficients $a_n$ grow or shrink roughly like $\\rho^n$, where $\\rho$ is the limit superior of $|a_n|^{1/n}$.\nThen $a_n x^n$ behaves like $(\\rho x)^n$, which is small exactly when $|x| \u003c 1/\\rho$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
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    "aliases": [
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    "name": "Cauchy–Hadamard theorem",
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    "same_as": {
      "wikidata": "Q859122",
      "wikipedia": "Cauchy–Hadamard theorem"
    },
    "sources": [
      {
        "locator": "Theorem for one complex variable",
        "url": "https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem"
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  "markdown": "https://math2.tamnd.com/node/07E5MB1CA2.md",
  "msc": [
    "30B10"
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  "name": "Cauchy–Hadamard theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07E/07E5MB1CA2.md",
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