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  "body": "\n## Statement\n\nLet $\\sum a_n (x - c)^n$ be a real power series with $a_n \\ne 0$ for all large $n$.\nAssume that $|a_{n+1}| / |a_n| \\to L$ as $n \\to \\infty$, for some $L$ in $[0, \\infty]$.\nThen the [[radius of convergence|0660QX1VPR]] of the series is $R = 1/L$, with the conventions $1/0 = \\infty$ and $1/\\infty = 0$.\n\n## Remarks\n\nThe proof applies the [[ratio test|0QNY6ZBXPN]] to $\\sum a_n (x - c)^n$. The ratio of consecutive terms tends to $L |x - c|$.\nThe formula needs the limit to exist. The [[Cauchy–Hadamard theorem|07E5MB1CA2]] works without this assumption.\nFor example, $\\sum x^n / n!$ has $L = 0$ and $R = \\infty$.\nMathlib states a form for nonzero $L$ as `FormalMultilinearSeries.ofScalars_radius_eq_inv_of_tendsto`.\n\n## Explanations\n\n### L1 human\n\nWhen the ratio of consecutive coefficients settles to $L$, the coefficients behave like $L^n$.\nSo the series behaves like a geometric series with ratio $L |x - c|$.\n",
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