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  "body": "\n## Definition\n\nLet $\\sum_{n \\ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with center $c \\in \\mathbb{C}$ and complex coefficients $a_n$.\nIts radius of convergence is\n$$R = \\sup \\{ r \\ge 0 : \\text{the sequence } (|a_n| r^n)_{n \\ge 0} \\text{ is bounded} \\},$$\nan element of $[0, \\infty]$.\nIts disc of convergence is the open disc $D = \\{ z \\in \\mathbb{C} : |z - c| \u003c R \\}$. It is empty if $R = 0$, and it is $\\mathbb{C}$ if $R = \\infty$.\n\n## Remarks\n\nThe series converges absolutely at each $z \\in D$, and uniformly on each closed disc $|z - c| \\le r$ with $r \u003c R$. It diverges at each $z$ with $|z - c| \u003e R$.\nThe Cauchy–Hadamard formula gives $1/R = \\limsup_{n \\to \\infty} |a_n|^{1/n}$, with $1/0 = \\infty$ and $1/\\infty = 0$.\nOn the circle $|z - c| = R$, the series can converge at all points, at some points, or at no point.\nFor example, $\\sum z^n$, $\\sum z^n / n$ and $\\sum z^n / n^2$ all have $R = 1$. On $|z| = 1$, the first converges nowhere, the second everywhere except at $z = 1$, and the third everywhere.\n\n## Explanations\n\n### L1 human\n\nA power series converges inside a disc around its center and diverges outside the closed disc.\nThe radius of this disc is the radius of convergence. On the boundary circle, each case is possible.\n",
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