Sums of Powers by L'Hopital's Rule

Eduardo Dueñez; Asimina S. Hamakiotes; Steven J. Miller

paper

Sums of Powers by L'Hopital's Rule

arXiv:2302.03624

Abstract

For a positive integer $d$ , let $p_{d} (n) := 0^{d} + 1^{d} + 2^{d} + \dots + n^{d}$ ; i.e., $p_{d} (n)$ is the sum of the first $d^{th}$ -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.