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  "body": "\n## Statement\n\nLet $\\sum_{\\alpha \\in \\N^n} c_\\alpha z^\\alpha$ be a [[power series in several variables|R3QTCPTJK7]] with centre $0$.\nLet $w \\in \\C^n$ and $M \\ge 0$ be such that $\\abs{c_\\alpha w^\\alpha} \\le M$ for every $\\alpha \\in \\N^n$.\nLet $0 \u003c \\rho \u003c 1$ and $K_\\rho = \\set{z \\in \\C^n : \\abs{z_j} \\le \\rho \\abs{w_j} \\text{ for } j = 1, \\dots, n}$.\nThen $\\sum_\\alpha \\abs{c_\\alpha z^\\alpha} \\le M (1 - \\rho)^{-n}$ for every $z \\in K_\\rho$, and the series converges absolutely and [[uniformly|08VZ57PT26]] on $K_\\rho$.\n\n## Remarks\n\nFor $z \\in K_\\rho$ one has $\\abs{c_\\alpha z^\\alpha} \\le \\abs{c_\\alpha w^\\alpha} \\rho^{\\abs{\\alpha}} \\le M \\rho^{\\abs{\\alpha}}$, and $\\sum_{\\alpha \\in \\N^n} \\rho^{\\abs{\\alpha}} = (1 - \\rho)^{-n}$ is a product of $n$ geometric series; the Weierstrass M-test concludes.\nIf all $w_j \\ne 0$, every compact subset of the [[polydisc|DJBWP7D3M8]] $\\set{\\abs{z_j} \u003c \\abs{w_j}}$ lies in some $K_\\rho$, so the series converges absolutely there, uniformly on compact subsets.\nIn particular, convergence at one point $w$ with all $w_j \\ne 0$ already gives convergence on this whole open polydisc.\nExample: for $\\sum_{k \\ge 0} (z_1 z_2)^k$ and $w = (2, 1/2)$, all terms have modulus $1$, so the series converges on $\\set{\\abs{z_1} \u003c 2, \\abs{z_2} \u003c 1/2}$.\nThe same series also converges on $\\set{\\abs{z_1} \u003c 1/2, \\abs{z_2} \u003c 2}$, so its domain of convergence $\\set{\\abs{z_1 z_2} \u003c 1}$ is larger than any single polydisc given by the lemma.\n\n## Explanations\n\n### L1 human\n\nIf the terms of a power series stay bounded at one point, the series converges at every point that is strictly smaller in each coordinate.\nThe reason is that the terms are then dominated by a convergent geometric series.\nThis is the same idea as for power series in one variable, done coordinate by coordinate.\n\n### L2 human\n\nThe lemma is the first step in describing where a power series converges.\nIt shows that the domain of convergence is a union of polydiscs centred at $0$, hence a complete Reinhardt domain.\nCombined with Hölder's inequality it gives the [[logarithmic convexity of the domain of convergence|D8S8JWE45R]].\nIt also shows that the sum is a uniform limit of polynomials on compact sets, hence holomorphic.\n",
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  "name": "Abel's lemma for multiple power series",
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