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  "body": "\n## Statement\n\nThe [[radius of convergence|0660QX1VPR]] of the power series $\\sum_{n=0}^{\\infty} \\frac{x^n}{n!}$ is $\\infty$.\nSo the series converges absolutely for every real $x$.\n\n## Remarks\n\nThe ratio of consecutive coefficients is $\\frac{1/(n+1)!}{1/n!} = \\frac{1}{n + 1} \\to 0$. The [[ratio formula|0JBMYAA9QH]] then gives radius $\\infty$.\nThe same holds for the series of sine, cosine, $\\sinh$ and $\\cosh$.\nMathlib states it as `NormedSpace.expSeries_radius_eq_top`.\n\n## Explanations\n\n### L1 human\n\nThe factorials in the denominators grow faster than every power $x^n$.\nSo the terms of the series go to $0$ fast, for each fixed $x$.\n",
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