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  "body": "\n## Summary\n\nAbelian and Tauberian theorems are two opposite kinds of results about [[summability methods|8X25Q4XYMN]].\nAn Abelian theorem says that a method gives the usual value to every convergent series.\nMore generally, it says that summability by one method implies summability by another, with the same value.\n[[Abel's theorem|0S4EMDQS2F]] is the model.\nA Tauberian theorem is a partial converse.\nIt says that a summable series converges, provided its terms satisfy an extra [[Tauberian condition|6ZDP261K7R]].\n[[Tauber's theorem|8SK3GQ4S0M]] is the model.\nTauberian theorems for integral transforms, such as the [[Wiener–Ikehara theorem|KQ57DC3AKM]], give short proofs of the [[prime number theorem|0EHBY5Y84N]].\n\n## Explanations\n\n### L1 human\n\nSummation methods are ways to give a value to a series that may diverge.\nAn Abelian theorem checks that a method does no harm: a series that already has a sum keeps the same sum.\nA Tauberian theorem goes the other way: if a method gives a value and the terms are small enough, the series really converges to that value.\nFor example, if $\\sum a_n x^n \\to s$ as $x \\to 1^-$ and $n a_n \\to 0$, then $\\sum a_n = s$.\n",
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