---
tag: 141Y4NMGY7
kind: concept
name: Euler–Boole summation
msc: ["65B15", "11B68", "40A25"]
sources:
  - url: https://en.wikipedia.org/wiki/Euler–Boole_summation
same_as:
  wikidata: Q30676419
  wikipedia: 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|9P839VB8S7]] 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.

## Explanations

### L1 human

The Euler–Maclaurin formula compares a sum $f(0) + f(1) + \cdots$ with an integral of $f$.
For an alternating sum $f(0) - f(1) + f(2) - \cdots$, the terms nearly cancel in pairs, and no integral is needed.
What remains is controlled by the ends of the sum: the values of $f$ and its derivatives there.
Euler–Boole summation writes this down exactly, with an error term that is small when high derivatives of $f$ are small.