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  "body": "\n## Definition\n\nLet $\\sum_{n \\ge 0} a_n$ be a [[series|0JH4MN4K1C]] of complex numbers.\nThe series is *Abel summable* to $s$ if the following two conditions hold.\n\n1. The [[power series|0VP56B4EMX]] $f(x) = \\sum_{n \\ge 0} a_n x^n$ converges for every real $x$ with $0 \\le x \u003c 1$.\n2. The [[one-sided limit|0909PWZR62]] $\\lim_{x \\to 1^-} f(x)$ exists and equals $s$.\n\n## Remarks\n\nExample: for $a_n = (-1)^n (n + 1)$, that is $1 - 2 + 3 - 4 + \\cdots$, one has $f(x) = \\sum_{n \\ge 0} (-1)^n (n+1) x^n = \\frac{1}{(1 + x)^2}$ for $0 \\le x \u003c 1$.\nThis tends to $\\frac14$ as $x \\to 1^-$, so the series is Abel summable to $1/4$.\n\nNon-example: for $1 + 1 + 1 + \\cdots$, condition 1 holds with $f(x) = \\frac{1}{1 - x}$.\nBut $f(x) \\to +\\infty$ as $x \\to 1^-$, so condition 2 fails and the series is not Abel summable.\n\nIf $\\sum a_n$ converges to $s$, then it is Abel summable to $s$; this is Abel's theorem on power series.\nIn the notation of summability, one writes $A\\text{-}\\sum a_n = s$.\n",
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