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  "body": "\n## Statement\n\nLet $\\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.\nLet $\\sigma \\colon \\mathbb{N} \\to \\mathbb{N}$ be a bijection.\nThen the [[rearrangement|0HRJ2Y74T3]] $\\sum a_{\\sigma(n)}$ converges absolutely, and its sum is $A$.\n\n## Remarks\n\nGiven $\\varepsilon \u003e 0$, choose $N$ with $\\sum_{n \u003e N} |a_n| \u003c \\varepsilon$. For large $m$, the first $m$ terms of the rearrangement include $a_0, \\dots, a_N$. So the two partial sums differ by less than $\\varepsilon$.\nBy the [[Riemann rearrangement theorem|0E7VBYGFF7]], the converse also holds for real series.\nMathlib states a general form for sums over a type, as `Equiv.tsum_eq`.\n\n## Explanations\n\n### L1 human\n\nWhen the series converges absolutely, the far terms have a small total size, whatever their order.\nSo a new order changes the partial sums only by a small amount.\n",
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