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  "body": "\n## Definition\n\nLet $M$ be a [[summability method|8X25Q4XYMN]] that assigns values to some [[series|0JH4MN4K1C]] of complex numbers.\n\n1. An *Abelian theorem* for $M$ is a theorem stating that every [[convergent series|0CVVKY7GY2]] is $M$-summable to its sum.\nMore generally, an Abelian theorem for two methods $M$ and $M'$ states that every $M$-summable series is $M'$-summable to the same value.\n2. A *Tauberian theorem* for $M$ is a theorem stating that an $M$-summable series converges if its terms satisfy a stated additional condition.\nIts sum is then its $M$-value.\nThe additional condition is called a [[Tauberian condition|6ZDP261K7R]].\n\nThe same terms are used for methods that act on sequences, functions or measures, with \"convergent\" replaced by the corresponding ordinary limit.\n\n## Remarks\n\nExample of an Abelian theorem: Abel's theorem on power series says that if $\\sum a_n$ converges to $s$, then $\\sum a_n x^n \\to s$ as $x \\to 1^-$.\nExample of a Tauberian theorem: Tauber's theorem says that if $\\sum a_n x^n \\to s$ as $x \\to 1^-$ and $n a_n \\to 0$, then $\\sum a_n$ converges to $s$.\n\nNon-example: the statement \"if $\\sum a_n x^n \\to s$ as $x \\to 1^-$, then $\\sum a_n$ converges\" has no Tauberian condition, and it is false.\nFor $a_n = (-1)^n$ one has $\\sum a_n x^n = \\frac{1}{1+x} \\to \\frac12$, but $\\sum (-1)^n$ diverges.\n\nThe names honour Abel and Tauber.\n",
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