Sums of Powers and the Bernoulli Numbers

TL;DRAbstract

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Ma

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