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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 0}$ be complex numbers such that $\\sum_{n \\ge 0} a_n x^n$ converges for $0 \\le x \u003c 1$.\nAssume that the series $\\sum a_n$ is [[Abel summable|EZXPY4A19C]] to $s$, that is, $\\sum_{n \\ge 0} a_n x^n \\to s$ as $x \\to 1^-$.\nAssume also that $a_n = O(1/n)$ in [[big O notation|0RYTGBJAGN]].\nThen $\\sum_{n \\ge 0} a_n$ [[converges|0CVVKY7GY2]] and its sum is $s$.\n\n## Remarks\n\nLittlewood's original proof was long.\nKaramata found a short proof.\nIt first controls the sums $\\sum a_n g(x^n)$ for polynomials $g$ and then approximates the indicator function of an interval from above and below by polynomials, which isolates the partial sum $s_N$.\nThe same method proves the [[Hardy–Littlewood Tauberian theorem|N42JDBT0VA]].\n\nThe Tauberian condition cannot be dropped.\nFor $a_n = (-1)^n$, $\\sum a_n x^n = \\frac{1}{1+x} \\to \\frac12$, but $\\sum (-1)^n$ diverges; here $n a_n$ is unbounded.\n\n## Explanations\n\n### L1 human\n\nAbel summation damps the $n$-th term by $x^n$ and lets $x \\to 1$.\nIt can give a value to a divergent series.\nLittlewood's theorem says that this cannot happen when the terms shrink at least like $1/n$: then the Abel value is the true sum.\n\n### L2 human\n\nThe theorem strengthens [[Tauber's theorem|8SK3GQ4S0M]], where the condition is $n a_n \\to 0$, and it is the Abel analogue of [[Hardy's Tauberian theorem for Cesàro means|BWWZB1F0HY]].\nBecause Abel summation is stronger than Cesàro summation, it implies Hardy's theorem.\nLittlewood's theorem was a starting point of the systematic study of Tauberian theorems by Hardy and Littlewood.\n",
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