Much of the appeal of the theory of numbers lies in the strikingly elegant and surprising methods that arise in the solution of its problems. Proof and method are of paramount concern in the subject. Studying the subject comes down to learning about the classic methods that have already been used. However, one problem after another arises that seems to require totally new methods. Historically, many of the standard techniques discussed in courses on calculus and analysis were first conceived to solve problems in number theory. Below we present some examples of how Taylor series expansions are used to study problems that first appear only to involve integers.