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  "body": "\n## Statement\n\nLet $\\sum_{n \\ge 0} a_n$ be a series of complex numbers that is [[Cesàro summable|N8ZGG7EW6J]] to $s$, that is, $\\sigma_n = \\frac{s_0 + \\cdots + s_n}{n + 1} \\to s$, where $s_n = a_0 + \\cdots + a_n$.\nThen the [[power series|0VP56B4EMX]] $f(x) = \\sum_{n \\ge 0} a_n x^n$ converges for $\\abs{x} \u003c 1$, and $f(x) \\to s$ as $x \\to 1^-$.\nSo the series is [[Abel summable|EZXPY4A19C]] to $s$.\n\n## Remarks\n\nThe sequence $(\\sigma_n)$ converges, so it is bounded, and then $s_n = (n+1)\\sigma_n - n\\sigma_{n-1}$ and $a_n = s_n - s_{n-1}$ are $O(n)$; hence the power series converges for $\\abs{x} \u003c 1$.\nMultiplying by $\\frac{1}{1-x}$ twice gives, for $\\abs{x} \u003c 1$,\n$$f(x) = (1 - x)^2 \\sum_{n \\ge 0} (n + 1) \\sigma_n x^n.$$\nSince $(1 - x)^2 \\sum_{n \\ge 0} (n+1) x^n = 1$, one gets $f(x) - s = (1 - x)^2 \\sum_{n \\ge 0} (n + 1)(\\sigma_n - s) x^n$.\nSplitting the sum at a large $N$ where $\\abs{\\sigma_n - s} \u003c \\varepsilon$ for $n \u003e N$ shows that this tends to $0$ as $x \\to 1^-$.\n\nExample: $1 - 1 + 1 - \\cdots$ is Cesàro summable to $\\frac12$, and indeed $\\sum (-1)^n x^n = \\frac{1}{1+x} \\to \\frac12$.\n\nThe converse fails: $1 - 2 + 3 - 4 + \\cdots$ is Abel summable to $\\frac14$, since $\\sum (-1)^n (n+1) x^n = \\frac{1}{(1+x)^2}$, but it is not Cesàro summable of order $1$.\n\n## Explanations\n\n### L1 human\n\nCesàro summation averages the partial sums; Abel summation damps the terms by powers of $x$ and lets $x \\to 1$.\nThe theorem says that the second method is at least as strong as the first and gives the same answer.\nEvery series that averaging can sum, damping can sum too, to the same value.\n\n### L2 human\n\nThis is an Abelian theorem in the sense of summability: it compares two methods and shows an inclusion with consistency.\nIt is a strengthening of [[Abel's theorem|0S4EMDQS2F]], because every convergent series is Cesàro summable to its sum.\nThe same argument shows that Cesàro summability of any order $k$ implies Abel summability to the same value.\n",
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  "name": "Frobenius theorem on Cesàro and Abel summability",
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