Elementary Properties

Abstract

We begin with power series on the real line ℝ. A formal expression

$$\sum\limits_{j = 0}^\infty {{a_j}{{(x - \alpha )}^j}} $$

, with the a _j ’s being either real or complex constants, is called a power series. It is usually convenient to take the coefficients a _i to all be real; there is no loss of generality in doing so. Our first task is to determine the nature of the set on which a power series converges.