Introduction to Power Series

Special power series were encountered in Sec. 2,7 in connection with the Cauchy product of series. In general, a power series in C is a series of functions ∑ m ∞ f k , m ≥ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003209072/1cd9098a-1579-4bfe-ada4-8961c2c40192/content/inline-math396.tif"/> , where f_k(z) = a _k (z — a)^k with a, a_k ∊ C, k = m, m + 1, m + 2, . . . . The numbers a_k , k = m, m + 1, . . ., are called the coefficients of the series. In the case m = 0, we use the convention (z — a)⁰ = 1 even when z = a.