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  "body": "\n## Definition\n\nLet $\\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].\nIts radius of convergence is\n$$R = \\sup \\{ r \\ge 0 : \\text{the sequence } (|a_n| r^n) \\text{ is bounded} \\},$$\nan element of $[0, \\infty]$.\n\n## Remarks\n\nThe series converges absolutely for $|x - c| \u003c R$, and it diverges for $|x - c| \u003e R$.\nAt the points $x = c \\pm R$, the series may converge or diverge.\nMany texts define $R$ as the supremum of all $|x - c|$ where the series converges. The two definitions give the same value.\nThe [[Cauchy–Hadamard theorem|07E5MB1CA2]] gives a formula for $R$. Mathlib uses the bounded form, in `FormalMultilinearSeries.radius`.\n\n## Explanations\n\n### L1 human\n\nA power series converges on an interval centered at $c$, and diverges outside it.\nThe radius of convergence is the half length of this interval.\n",
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