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  "body": "\n## Definition\n\nLet $M$ be a [[summability method|8X25Q4XYMN]] that assigns a value to some [[series|0JH4MN4K1C]] $\\sum a_n$ of real or complex numbers.\nLet $P$ be a property of sequences $(a_n)_{n \\ge 0}$.\nThe property $P$ is a *Tauberian condition* for $M$ if the following holds.\nEvery series $\\sum a_n$ whose terms have property $P$ and which is $M$-summable to $s$ [[converges|0CVVKY7GY2]] with sum $s$.\n\n## Remarks\n\nExample: for Abel summation, the property $n a_n \\to 0$ is a Tauberian condition.\nThis is [[Tauber's theorem|8SK3GQ4S0M]].\nStronger theorems allow weaker conditions; for example, $n a_n = O(1)$ is a Tauberian condition both for Cesàro summation and for Abel summation.\n\nNon-example: the property $a_n \\to 0$ is not a Tauberian condition for Cesàro summation.\nLet the partial sums start at $-1$.\nFor $k = 1, 2, 3, \\dots$ in turn, let them run up to $1$ in $k$ steps of size $2/k$ and then back down to $-1$ in $k$ such steps.\nThe terms are $\\pm 2/k$, so $a_n \\to 0$.\nOver each full up-and-down run the partial sums add up to $0$.\nThe incomplete last run contributes at most $2k + 2$ in absolute value, while $n$ is at least of order $k^2$.\nSo the arithmetic means of the partial sums tend to $0$, but the partial sums reach $1$ and $-1$ infinitely often and do not converge.\n\nA theorem whose hypothesis is a Tauberian condition is called a Tauberian theorem.\nIt is a partial converse of an Abelian theorem such as [[Abel's theorem|0S4EMDQS2F]].\n",
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