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  "body": "\n## Statement\n\nLet $(V, \\|\\cdot\\|)$ be a [[Banach space|0BDTTM24DP]] and let $(x_n)_{n \\ge 1}$ be a sequence in $V$ with $\\sum_{n \\ge 1} \\|x_n\\| \u003c \\infty$.\nThen the partial sums $s_N = \\sum_{n=1}^{N} x_n$ converge to a point $s \\in V$, and $\\|s\\| \\le \\sum_{n \\ge 1} \\|x_n\\|$.\n\n## Remarks\n\nProof: for $M \u003e N$, the triangle inequality gives $\\|s_M - s_N\\| \\le \\sum_{n=N+1}^{M} \\|x_n\\|$. This tends to $0$, so $(s_N)$ is a [[Cauchy sequence|0MCK8K2D7Z]], and it converges because $V$ is [[complete|08EWT7J0SC]].\nThe converse holds: a normed space in which each absolutely convergent series converges is a Banach space.\nFor $V = \\mathbb{R}$, this is the fact that absolutely convergent real series converge.\n\n## Explanations\n\n### L1 human\n\nIf the lengths of the terms have a finite sum, then the series converges in a Banach space.\nThis is a practical test for completeness, and it is often used to build solutions as sums of series.\n",
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        "url": "https://en.wikipedia.org/wiki/Absolute_convergence"
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  "name": "Absolutely convergent series in a Banach space converge",
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