# David Rohrlich

## Course Description

We will start with the notion of a formal power series. While it is important that identities like

and

are equalities of real-valued functions, it is also important to be able to treat such equations as purely algebraic identities. Doing so allows one to prove, for example, that , this identity being a simple consequence of the binomial theorem.

Next we will introduce the Bernoulli numbers . They are dened by the identity of formal power series

We will also consider the Bernoulli polynomials , which are dened by

The Bernoulli numbers and Bernoulli polynomials have many remarkable proper- ties. For example, the nite sum

has a simple expression in terms of Bernoulli polynomials, and the innite sum

has a simple expression in terms of Bernoulli numbers when is even. The latter sum is actually the value at of the Riemann zeta function , a function which is at the heart of the most famous unsolved problem in mathematics.