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  "body": "\n## Statement\n\nLet $(a_{m,n})_{m, n \\in \\N}$ be a [[double sequence|WK5V372TG3]] of complex numbers with\n$$\\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} \\abs{a_{m,n}} \u003c \\infty.$$\nThen all the series below converge [[absolutely|0W3K3QPEWT]], and\n$$\\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} a_{m,n} = \\sum_{n=0}^{\\infty} \\sum_{m=0}^{\\infty} a_{m,n}.$$\n\n## Remarks\n\nBy the nonnegative case of [[Fubini's theorem for series|WW53KG71KH]], the hypothesis says that the family $(a_{m,n})$ over $\\N \\times \\N$ is summable.\nThe same theorem, applied once with rows first and once with columns first, shows that both iterated sums equal the sum of the family.\nThe hypothesis cannot be dropped.\nLet $a_{m,m} = 1$, $a_{m,m+1} = -1$ for all $m$, and $a_{m,n} = 0$ otherwise.\nEach row sums to $0$, so $\\sum_m \\sum_n a_{m,n} = 0$.\nColumn $0$ sums to $1$ and every other column sums to $1 - 1 = 0$, so $\\sum_n \\sum_m a_{m,n} = 1$.\nExample of use: with the [[Riemann zeta function|1XWBY4N0VT]], $\\sum_{k \\ge 2} (\\zeta(k) - 1) = \\sum_{k \\ge 2} \\sum_{n \\ge 2} n^{-k}$.\nSwapping the nonnegative sums gives $\\sum_{n \\ge 2} \\sum_{k \\ge 2} n^{-k} = \\sum_{n \\ge 2} \\frac{1}{n(n-1)} = 1$.\n\n## Explanations\n\n### L1 human\n\nFor an infinite table of numbers, summing rows first or columns first gives the same total when the absolute values have a finite total.\nWithout this condition, the two orders can give different answers.\n\n### L2 human\n\nThe result is used to swap two sums, which is one of the most common steps in analysis and number theory.\nTo apply it, bound the double sum of absolute values; for nonnegative terms the swap is always allowed, possibly with value $+\\infty$.\nThe counterexample above shows why conditionally convergent double sums need care.\n",
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