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  "body": "\n## Definition\n\nLet $(a_{m,n})_{m, n \\in \\N}$ be a [[double sequence|WK5V372TG3]] of [[complex numbers|09FTAET5Z4]].\nThe double series $\\sum_{m, n = 0}^{\\infty} a_{m,n}$ is the double sequence of rectangular partial sums\n$$S_{M,N} = \\sum_{m=0}^{M} \\sum_{n=0}^{N} a_{m,n}.$$\nThe double series converges to $s \\in \\C$ if $(S_{M,N})$ converges to $s$ in the sense of [[Pringsheim|QYTGCW1GTK]].\nThat is, for every $\\varepsilon \u003e 0$ there is $K$ with $\\abs{S_{M,N} - s} \u003c \\varepsilon$ for all $M, N \\ge K$.\n\n## Remarks\n\nExample: for $a_{m,n} = 2^{-m} 3^{-n}$ one has $S_{M,N} = \\left(\\sum_{m \\le M} 2^{-m}\\right)\\left(\\sum_{n \\le N} 3^{-n}\\right)$.\nThe two factors tend to $2$ and $\\tfrac32$, so the double series converges to $3$.\nNon-example: for $a_{m,n} = 1$ when $m = n$ and $0$ otherwise, $S_{M,N} = \\min(M, N) + 1$, which is unbounded, so the double series diverges.\nA convergent double series can have unbounded terms.\nWith $a_{0,n} = n$, $a_{1,n} = -n$ and $a_{m,n} = 0$ for $m \\ge 2$, one has $S_{M,N} = 0$ for all $M \\ge 1$.\nSo it converges to $0$, although its rows do not form convergent series.\n\n## Explanations\n\n### L1 human\n\nA double series is the sum of all entries of an infinite table, taken as the limit of sums over larger and larger rectangles.\n",
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