Prove that a power series converges absolutely everywhere or nowhere on its circle of convergence.
Every complex power series has a certain radius of convergence, inside of which the series is guaranteed to converge. What happens on the very edge of this region, when the radius is equal to the radius of convergence? In general, we cannot say, but it is true that the series either converges absolutely everywhere or nowhere on the circle of convergence. The solution comprises a one page Word attachment, with equations written in Mathtype, proving this fact, and also giving an example of each case.