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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $M$ be a [[summability method|X80DZEXQDQ]].\nWrite $M\\left(\\sum a_n\\right)$ for the $M$-sum of an $M$-summable series $\\sum_{n \\ge 0} a_n$ with terms in $\\mathbb{K}$.\n\n1. $M$ is *linear* if, whenever $\\sum a_n$ and $\\sum b_n$ are $M$-summable and $\\lambda, \\mu \\in \\mathbb{K}$, the series $\\sum (\\lambda a_n + \\mu b_n)$ is $M$-summable and $M\\left(\\sum (\\lambda a_n + \\mu b_n)\\right) = \\lambda M\\left(\\sum a_n\\right) + \\mu M\\left(\\sum b_n\\right)$.\n2. $M$ is *stable* if, for every sequence $(a_n)_{n \\ge 0}$, the series $a_0 + a_1 + a_2 + \\cdots$ is $M$-summable if and only if $a_1 + a_2 + \\cdots$ is $M$-summable, and then $M(a_0 + a_1 + a_2 + \\cdots) = a_0 + M(a_1 + a_2 + \\cdots)$.\n\n## Remarks\n\nExample: ordinary summation of convergent series is linear and stable, by the limit rules for sums and because shifting the partial sums by $a_0$ does not affect convergence.\nCesàro summation is also linear and stable.\nNon-example: zeta function regularization assigns $1 + 1 + 1 + \\cdots$ the value $\\zeta(0) = -\\frac12$; stability would give $x = 1 + x$ for this value, so this method is not stable.\nIn terms of the partial sums $s_n$, linearity says that $D_M$ is a vector space and $M$ is a linear map on it.\n\n## Explanations\n\n### L1 human\n\nLinear means sums and multiples are respected; stable means the first term can be split off.\n",
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