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  "body": "\n## Summary\n\nA series $\\sum x_n$ of vectors in a [[Banach space|06GZ956657]] converges when its partial sums converge in norm.\nIt converges absolutely when the real series $\\sum \\norm{x_n}$ converges.\n[[Absolute convergence implies convergence|082MGNWNA8]], because the space is complete.\nIn infinite dimensions the converse fails more sharply than in $\\R$: there are series that converge after every rearrangement but not absolutely.\nSeries of this kind are used everywhere in analysis, for example in the Neumann series $\\sum T^n$ for an operator $T$ of norm less than $1$.\n\n## Explanations\n\n### L1 human\n\nA series of vectors is added term by term, like a series of numbers, and one asks whether the partial sums approach a vector.\nThe size of a vector is its norm, so convergence means that the norm of the difference between the partial sums and the limit tends to $0$.\nIf the sizes $\\norm{x_n}$ add up to a finite number, the series converges, just as for numbers.\nFor example, in the space $\\ell^2$ of square-summable sequences, with $e_n$ the $n$-th unit vector, the series $e_1 + e_2/2 + e_3/3 + \\cdots$ can converge even though $1 + 1/2 + 1/3 + \\cdots$ diverges.\n",
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