On the sum of the r -th powers of the first n integers

Abstract

In this note an explicit expression is obtained for the sum of the r-th powers of the first n integers. The result is equivalent to the well-known result in terms of Bernoulli numbers and the equivalence is not difficult to establish. However, the method given here is elementary and self-contained and provides an excellent exercise on. the manipulation of determinants. Throughout the note, the following notation will be used: S r s l ' + 2 ' + . . . + n, n f\=n(n-l)(n-2)...(n-r+