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  "body": "\n## Definition\n\nLet $n \\ge 0$ and let $\\C[\\![z_1, \\dots, z_n]\\!]$ be the ring of [[formal power series|KCBFS516DE]] in $n$ variables over $\\C$.\nFor a multi-index $\\alpha = (\\alpha_1, \\dots, \\alpha_n) \\in \\N^n$ write $z^\\alpha = z_1^{\\alpha_1} \\cdots z_n^{\\alpha_n}$ and $\\abs{\\alpha} = \\alpha_1 + \\cdots + \\alpha_n$.\nA [[power series|R3QTCPTJK7]] $f = \\sum_{\\alpha} c_\\alpha z^\\alpha$ is *convergent* if there is a real number $r \u003e 0$ with\n$$\\sum_{\\alpha \\in \\N^n} \\abs{c_\\alpha}\\, r^{\\abs{\\alpha}} \u003c \\infty.$$\nThe *ring of convergent power series* $\\C\\{z_1, \\dots, z_n\\}$ is the set of convergent power series, with the addition and multiplication of formal power series.\n\n## Remarks\n\nThe condition says that the series converges absolutely on the closed polydisc of radius $r$ about $0$; it is equivalent to convergence on some neighbourhood of $0$.\nSums and products of convergent series are convergent, so $\\C\\{z_1, \\dots, z_n\\}$ is a subring of $\\C[\\![z_1, \\dots, z_n]\\!]$.\nExample: $f = \\sum_{k \\ge 0} z_1^k z_2^k$ is convergent: with $r = 1/2$, $\\sum_k r^{2k} = 4/3 \u003c \\infty$.\nNon-example: $g = \\sum_{k \\ge 0} k!\\, z_1^k$ is not convergent, since $k!\\, r^k \\to \\infty$ for every $r \u003e 0$.\nFor $n = 0$ the ring is $\\C$.\nA convergent series is a unit if and only if its constant term is nonzero, so the ring is local with maximal ideal $(z_1, \\dots, z_n)$.\nSending a series to its sum near $0$ gives an isomorphism with the ring $\\mathcal{O}_{n,0}$ of germs of holomorphic functions at $0$.\n",
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