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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $\\mathbb{K}^{\\N}$ be the set of all [[sequences|0QE8STGJH2]] in $\\mathbb{K}$.\nA *summability method* $M$ consists of a set $D_M \\subseteq \\mathbb{K}^{\\N}$ and a map $M \\colon D_M \\to \\mathbb{K}$.\nA sequence $s$ is *$M$-summable* (or *$M$-limitable*) to $M(s)$ if $s \\in D_M$.\nA [[series|0JH4MN4K1C]] $\\sum a_n$ is $M$-summable with $M$-sum $L$ if its sequence of [[partial sums|0NB643K60Q]] $s_n = a_0 + \\cdots + a_n$ is $M$-summable to $L$.\n\n## Remarks\n\nExample: the ordinary limit, with $D_M$ the set of convergent sequences and $M(s) = \\lim s_n$, is a summability method; it sums exactly the convergent series.\nExample: Cesàro summation takes $D_M$ to be the sequences whose [[Cesàro means|0P0DJHBGBB]] converge, and $M(s)$ the limit of these means; it sums $1 - 1 + 1 - \\cdots$ to $\\frac12$.\nNon-example: \"take the limit of some convergent subsequence\" is not a summability method.\nThe sequence $0, 1, 0, 1, \\dots$ would get both $0$ and $1$, so the rule does not define a map.\nSome authors define methods directly on series $(a_n)$ instead of on partial sums; the two points of view are equivalent.\n\n## Explanations\n\n### L1 human\n\nA summability method is any well-defined rule that turns certain sequences into numbers.\nFurther conditions, such as agreement with the ordinary limit, single out the useful methods.\n",
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