ABSTRACT
When people learn about power series, and especially Taylor's formula, in calculus class people generally come away with the impression that most any function can be expanded in a power series. Unfortunately this is not the case. Functions that have convergent power series expansions are called real analytic functions, and have many special properties. This chapter discusses Hadamard's elegant formula for the radius of convergence of a power series, and learnt more about the Taylor expansion. The single most important attribute of a power series is its radius of convergence. It turns out that the radius of convergence is best understood in the context of the complex numbers. The chapter gives rigorous definitions of the exponential and trigonometric functions, and derives some of their most basic properties. The logarithm function is of interest because it is the inverse of the exponential function, but also because it is used to define entropy in physics.
