#### Degree Name

Master of Arts (MA)

#### Semester of Degree Completion

1996

#### Thesis Director

Duane M. Broline

#### Abstract

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* /*e ^{z }*- 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of

*f*(

*z*)

*,*and the zeta function. Further areas of research relating to the topic are explored.

#### Recommended Citation

Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). *Masters Theses*. 1896.

https://thekeep.eiu.edu/theses/1896