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  "body": "\n## Definition\n\nLet $M \\colon D_M \\to \\mathbb{K}$ be a [[summability method|X80DZEXQDQ]], where $\\mathbb{K}$ is $\\R$ or $\\C$.\nThe method $M$ is *regular* if every [[convergent sequence|0AZ3R52593]] $s$ belongs to $D_M$ and\n$$M(s) = \\lim_{n \\to \\infty} s_n.$$\nEquivalently, every convergent series is $M$-summable, and its $M$-sum is its ordinary sum.\n\n## Remarks\n\nExample: Cesàro summation is regular, because the [[Cesàro means of a convergent sequence converge to the same limit|0RDRPBMNAC]].\nNon-example: the method with $D_M$ the convergent sequences and $M(s) = 2 \\lim s_n$ is a summability method, but it is not regular, since it sends the constant sequence $1$ to $2$.\nA method that sums every convergent sequence, but not necessarily to its ordinary limit, is called *conservative*.\n\n## Explanations\n\n### L1 human\n\nA regular method extends the ordinary limit: it never changes the value of a convergent sequence.\n",
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