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  "body": "\n## Summary\n\nA power series in several variables is an infinite sum $\\sum_\\alpha c_\\alpha (z - a)^\\alpha$ of monomials in $z_1 - a_1, \\dots, z_n - a_n$, indexed by [[multi-indices|0PJXBR6QA3]] $\\alpha$.\nIt extends the one-variable [[power series|0MZB06ST2B]].\nA function on an open set of $\\C^n$ is [[holomorphic|507C34W0ZT]] exactly when it is locally the sum of a convergent power series.\nThe domain of convergence of such a series is the interior of the set where it converges.\nUnlike in one variable, it is not always a polydisc; it is a [[logarithmically convex|EHWH6EF4QY]] complete [[Reinhardt domain|WA0SE56FVH]].\n\n## Explanations\n\n### L1 human\n\nA polynomial in $z_1$ and $z_2$ is a finite sum of terms $c \\, z_1^j z_2^k$.\nA power series allows infinitely many such terms.\nFor example, $\\sum_{j, k \\ge 0} z_1^j z_2^k$ is the product of two geometric series, and it equals $\\frac{1}{(1 - z_1)(1 - z_2)}$ when $\\abs{z_1} \u003c 1$ and $\\abs{z_2} \u003c 1$.\nThe terms have no natural order, so one asks the sum of their absolute values to be finite.\n",
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