---
tag: 81Q4ERP594
kind: concept
name: Euler summation
msc: ["40G05"]
specializes: [8X25Q4XYMN]
sources:
  - url: https://en.wikipedia.org/wiki/Euler_summation
same_as:
  wikidata: Q374926
  wikipedia: Euler summation
---

## Summary

Euler summation is a [[summability method|8X25Q4XYMN]] that replaces a series by a new series built from binomial combinations of its terms.
The value is the sum of the new series.
It goes back to the [[Euler transform|WT0HACKQV7]] of alternating series, which speeds up their convergence.
The method gives the usual sum to every convergent series and sums many divergent alternating series.
It belongs to the family of Hausdorff methods, together with the Cesàro methods.

## Explanations

### L1 human

Euler summation mixes neighbouring terms of a series with binomial weights, as in the rows of Pascal's triangle.
For an alternating series, large terms of opposite sign then cancel each other.
For $1 - 1 + 1 - 1 + \cdots$, every mixed term after the first is $0$, and the new series is just $\frac12$.
So Euler summation gives the value $1/2$, as Cesàro and Abel summation do.