Sums of Powers by L'Hopital's Rule

TL;DR

This paper presents a novel proof that the sum of powers is a polynomial using L'Hopital's rule, offering an alternative to induction and illustrating the value of multiple mathematical approaches.

Contribution

It introduces a simple proof leveraging L'Hopital's rule to establish the polynomial nature of power sums, bypassing induction and demonstrating the utility of diverse methods.

Findings

Proof using L'Hopital's rule for polynomial degree of power sums

Method to determine coefficients via Cramer's rule

Highlights importance of multiple problem-solving techniques

Abstract

For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$; i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's well known that $p_d(n)$ is a polynomial of degree $d+1$ in $n$. While this is usually proved by induction, once $d$ is not small it's a challenge as one needs to know the polynomial for the inductive step. We show how this difficulty can be bypassed by giving a simple proof that $p_d(n)$ is a polynomial of degree $d+1$ in $n$ by using L'Hopital's rule, and show how we can then determine the coefficients by Cramer's rule. This illustrates a general principle and the point of our paper: there's more than one path to a goal, different approaches have their advantages and disadvantages, and the more techniques one knows, the more likely one can successfully attack a problem.