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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 0}$ be a sequence of complex numbers such that the [[power series|0VP56B4EMX]] $f(x) = \\sum_{n=0}^{\\infty} a_n x^n$ converges for $0 \\le x \u003c 1$.\nAssume that the [[one-sided limit|0909PWZR62]] $s = \\lim_{x \\to 1^-} f(x)$ exists.\nAssume also that $n a_n \\to 0$ as $n \\to \\infty$.\nThen the series $\\sum_{n=0}^{\\infty} a_n$ [[converges|0CVVKY7GY2]], and its sum is $s$.\n\n## Remarks\n\nThe idea of the proof is to compare the partial sum $s_N = \\sum_{n \\le N} a_n$ with $f(x)$ for $x = 1 - 1/N$.\nThe difference is at most $\\sum_{n \\le N} \\abs{a_n} (1 - x^n) + \\sum_{n \u003e N} \\abs{a_n} x^n$.\nUse $1 - x^n \\le n(1 - x)$ in the first sum, which is then at most the average of $n \\abs{a_n}$ over $n \\le N$, and this tends to $0$.\nIn the second sum, $\\abs{a_n} \\le \\varepsilon / n \\le \\varepsilon / N$ for large $N$, so it is at most $\\varepsilon / (N(1 - x)) = \\varepsilon$.\n\nThe theorem is a partial converse of [[Abel's theorem|0S4EMDQS2F]], which gives $\\lim_{x \\to 1^-} f(x) = \\sum a_n$ whenever the series converges.\nThe condition on $n a_n$ cannot be dropped.\nFor $a_n = (-1)^n$, one has $f(x) = 1/(1 + x) \\to 1/2$, but $\\sum (-1)^n$ diverges, and $n a_n$ does not tend to $0$.\nLittlewood showed that the weaker condition $n a_n = O(1)$ is enough; see [[big O notation|0RYTGBJAGN]].\n\n## Explanations\n\n### L1 human\n\nAbel's theorem says that a convergent series can be summed by plugging $x \\to 1$ into its power series.\nThe converse is false in general: the power series can have a limit even when the series diverges.\nTauber's theorem says that the converse does hold when the terms shrink faster than $1/n$.\n\n### L2 human\n\nThis is the first Tauberian theorem, and it gave its name to the whole family of [[Abelian and Tauberian theorems|D6XWK1NRY1]].\nAn Abelian theorem passes from ordinary convergence to convergence of a summation method; a Tauberian theorem goes back, under an extra size condition on the terms.\nLater results of this type, such as the Hardy–Littlewood and Wiener Tauberian theorems, are central tools in analytic number theory.\n",
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