The Sums of Integer Powers

Abstract

An investigation of the origin of the formulas for the sums of integer powers was performed. A method for calculating the sums of the first n integers to the kth power, denoted Sk(n), was first derived by Jacques Bernoulli in the late 1600’s. Through the discovery of formulas for the computation of integer powers, a numeric sequence arose. This sequence has become known as the Bernoulli numbers. Bernoulli simultaneously derived a recursive algorithm that can generate Bernoulli numbers. This recursive relationship was the subject of the first computer program in 1843. Utilizing a relationship that Bernoulli observed, it is possible to then calculate all terms in a coefficient matrix that enables the calculation of formulas for Sk(n). This coefficient matrix was generated using computer software with two different methods. Numerical analysis techniques were applied to each generated matrix, and the relative error between the two matrices was compared.

Department: Mathematical Sciences 

Faculty Mentor: Dr. Christian Ivanescu