The Relationship between Series \(\sum_{i=1}^{n}{i^m}\) and the Eulerian Numbers

Abstract

In this paper, I study the Eulerian numbers \((A(m,k))_{k=1}^{m}\) and prove the relationship between \(\sum_{i=1}^{n}{i^m}\) and \((A(m,k))_{k=1}^{m}\), to be \(\sum_{i=1}^{n}{i^m} = \sum_{k=1}^m A(m,k)\binom{m+k}{m+1}\).