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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|0BDTTM24DP]] over $\\R$ or $\\C$ with norm $\\norm{\\cdot}$, and let $(x_n)_{n \\ge 0}$ be a sequence in $X$.\nThe [[partial sums|0NB643K60Q]] of the series $\\sum_n x_n$ are $S_N = \\sum_{n=0}^{N} x_n$.\nThe series *converges* to $s \\in X$, written $\\sum_{n \\ge 0} x_n = s$, if $\\norm{S_N - s} \\to 0$ as $N \\to \\infty$.\nThe series *converges absolutely* if the real series $\\sum_{n \\ge 0} \\norm{x_n}$ [[converges|0CVVKY7GY2]].\n\n## Remarks\n\n[[An absolutely convergent series in a Banach space converges|082MGNWNA8]].\nMore generally, if $\\norm{x_n} \\le b_n$ for all $n$ and $\\sum b_n$ converges, then $\\sum x_n$ converges; in Mathlib this is `Summable.of_norm_bounded`.\nA normed space in which every absolutely convergent series converges is complete, so this property characterizes Banach spaces.\n\nExample: in $\\ell^2$, see [[spaces ℓ^p|08P6T2WH6R]], let $x_n = e_n / n$ for $n \\ge 1$, where $e_n$ is the $n$-th unit vector.\nThen $\\norm{S_N - s}_2 = \\left( \\sum_{n \u003e N} 1/n^2 \\right)^{1/2} \\to 0$ for $s = (1, 1/2, 1/3, \\dots)$, so the series converges.\n\nNon-example: the same series does not converge absolutely, because $\\sum_n \\norm{x_n}_2 = \\sum_n 1/n$ diverges.\nThis series even converges after every rearrangement.\nIn $\\R$ this cannot happen: a real series that converges after every rearrangement converges absolutely.\n",
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