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  "body": "\n## Summary\n\nA Tauberian condition is an extra hypothesis on the terms of a series that lets one pass from summability back to ordinary convergence.\n[[Abel's theorem|0S4EMDQS2F]] says that a convergent series is Abel summable to its sum, but the converse fails.\n[[Tauber's theorem|8SK3GQ4S0M]] shows that the converse holds when $n a_n \\to 0$, and this condition is the first example of a Tauberian condition.\nSuch conditions are the subject of [[Tauberian theory|D6XWK1NRY1]].\n\n## Explanations\n\n### L1 human\n\nA [[summability method|8X25Q4XYMN]] can give a value to a series that does not converge, such as $1 - 1 + 1 - 1 + \\cdots$.\nSo knowing that a series is summable does not tell you that it converges.\nA Tauberian condition is a promise that the terms are small or tame enough to rule out such wild cases.\nFor example, if the terms satisfy $n a_n \\to 0$ and the series is Abel summable, then it really converges.\nThe terms of $1 - 1 + 1 - \\cdots$ do not shrink at all, so they fail every such condition.\n",
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