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 $\inline&space;e^{x+y}=e^xe^y$, this identity being a simple consequence of the binomial theorem.

Next we will introduce the Bernoulli numbers $\inline&space;b_k$. They are de ned by the identity of formal power series

We will also consider the Bernoulli polynomials $\inline&space;B_k(x)$, which are de ned 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 in nite sum

has a simple expression in terms of Bernoulli numbers when $\inline&space;k$ is even. The latter sum is actually the value at $\inline&space;s=k$ of the Riemann zeta function $\inline&space;\zeta(s)$, a function which is at the heart of the most famous unsolved problem in mathematics.