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  "body": "\n## Summary\n\nAbel summation is the [[summability method|8X25Q4XYMN]] that assigns to a series $\\sum a_n$ the limit of the [[power series|0VP56B4EMX]] $\\sum a_n x^n$ as $x \\to 1$ from below.\nBy [[Abel's theorem|0S4EMDQS2F]], it gives the usual sum to every convergent series.\nIt sums every series that [[Cesàro summation|N8ZGG7EW6J]] sums, to the same value, and more.\nIts partial converses, such as [[Tauber's theorem|8SK3GQ4S0M]] and [[Littlewood's Tauberian theorem|Z35TD4ZZRD]], started the theory of Tauberian theorems.\n\n## Explanations\n\n### L1 human\n\nDamp the $n$-th term of the series by the factor $x^n$, with $x$ slightly below $1$.\nThe damped series often converges even when the original does not.\nThen let $x$ grow to $1$ and see whether the damped sums approach a limit.\nFor $1 - 1 + 1 - \\cdots$, the damped sum is $1 - x + x^2 - \\cdots = \\frac{1}{1+x}$, which tends to $1/2$.\n",
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