# Bernoulli and Euler numbers from divergent series

## TL;DR

This paper offers a straightforward proof of identities involving Bernoulli and Euler numbers using Abel sums of divergent series, emphasizing the role of linearity in summation methods.

## Contribution

It introduces a simple proof technique for classical identities of Bernoulli and Euler numbers via Abel summation of divergent series.

## Key findings

- Identifies identities relating Bernoulli and Euler numbers.
- Demonstrates the use of Abel sums for divergent series.
- Highlights the importance of linearity in summation methods.

## Abstract

The aim of this note is to provide a simple proof of some well-known identities and recurrences relating classical Bernoulli and Euler numbers by using the Abel sum of the divergent series $\sum_{n=0}^\infty (-1)^{n} (n+1)^k$, $k$ a positive integers. Special attention is placed on the fact that the numerical value of these sums is determined by the linearity of the summation method involved.