Euler–Boole summation

Summary

Euler–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. An integral remainder makes the formula exact. The coefficients are values of the Euler polynomials, in the way that the Euler–Maclaurin formula uses Bernoulli polynomials for ordinary sums. It is the alternating counterpart of Euler–Maclaurin summation. It gives exact formulas for alternating sums of powers and asymptotic expansions of alternating series.