A simple recursive representation of the Faulhaber series

TL;DR

This paper introduces a straightforward recursive method to compute Faulhaber series, which sum powers of integers, avoiding the traditional use of Bernoulli numbers or polynomials.

Contribution

It provides a new elementary recursive representation of Faulhaber series that simplifies calculations and understanding.

Findings

Recursive formula for Faulhaber series derived

Simplifies computation of power sums

Eliminates need for Bernoulli numbers

Abstract

We present a simple elementary recursive representation of the so called Faulhaber series $\sum_{k=1}^n k^N$ for integer $n$ and $N$, without reference to Bernoulli numbers or polynomials.