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  "body": "\n## Definition\n\nLet $c \\in \\mathbb{C}$ and let $(a_n)_{n \\ge 0}$ be a sequence of [[complex numbers|09FTAET5Z4]].\nThe power series with center $c$ and coefficients $a_n$ is the series\n$$\\sum_{n=0}^{\\infty} a_n (z - c)^n$$\nin the complex variable $z$.\nThe series converges at a point $z \\in \\mathbb{C}$ if the partial sums $\\sum_{n=0}^{N} a_n (z - c)^n$ have a limit in $\\mathbb{C}$ as $N \\to \\infty$. The sum of the series is the function that maps each such $z$ to this limit.\n\n## Remarks\n\nEach power series converges at $z = c$, with sum $a_0$.\nThe geometric series $\\sum_{n \\ge 0} z^n$ converges if and only if $|z| \u003c 1$, and then its sum is $1/(1 - z)$.\nThe set where the series converges contains an open disc and is contained in its closure, see [[disc of convergence|045EWAXY9P]].\nA [[formal power series|033WGN00BF]] is the same expression without any question of convergence.\n\n## Explanations\n\n### L1 human\n\nA power series is a polynomial with infinitely many terms.\nFor some values of $z$ the terms get small fast enough and the series has a sum, and for other values it does not.\nFor example, $\\sum z^n$ has a sum when $|z| \u003c 1$, but not when $|z| \\ge 1$, because then the terms do not go to $0$.\n",
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