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  "body": "\n## Summary\n\nEuler–Boole summation expresses an alternating sum $\\sum (-1)^j f(j + h)$ through the values of $f$ and its derivatives at the two ends of the range.\nAn integral remainder makes the formula exact.\nThe coefficients are values of the Euler polynomials, in the way that the [[Euler–Maclaurin formula|9P839VB8S7]] uses Bernoulli polynomials for ordinary sums.\nIt is the alternating counterpart of Euler–Maclaurin summation.\nIt gives exact formulas for alternating sums of powers and asymptotic expansions of alternating series.\n\n## Explanations\n\n### L1 human\n\nThe Euler–Maclaurin formula compares a sum $f(0) + f(1) + \\cdots$ with an integral of $f$.\nFor an alternating sum $f(0) - f(1) + f(2) - \\cdots$, the terms nearly cancel in pairs, and no integral is needed.\nWhat remains is controlled by the ends of the sum: the values of $f$ and its derivatives there.\nEuler–Boole summation writes this down exactly, with an error term that is small when high derivatives of $f$ are small.\n",
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