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  "body": "\n## Summary\n\nEuler summation is a [[summability method|8X25Q4XYMN]] that replaces a series by a new series built from binomial combinations of its terms.\nThe value is the sum of the new series.\nIt goes back to the [[Euler transform|WT0HACKQV7]] of alternating series, which speeds up their convergence.\nThe method gives the usual sum to every convergent series and sums many divergent alternating series.\nIt belongs to the family of Hausdorff methods, together with the Cesàro methods.\n\n## Explanations\n\n### L1 human\n\nEuler summation mixes neighbouring terms of a series with binomial weights, as in the rows of Pascal's triangle.\nFor an alternating series, large terms of opposite sign then cancel each other.\nFor $1 - 1 + 1 - 1 + \\cdots$, every mixed term after the first is $0$, and the new series is just $\\frac12$.\nSo Euler summation gives the value $1/2$, as Cesàro and Abel summation do.\n",
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