Complex Analysis: Theory and Applications · Problem 4.4.2

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · radius-of-convergence-boundary

Problem

Consider the series

\displaystyle\sum_{n=0}^{\infty} z^n

\displaystyle\sum_{n=1}^{\infty} \frac{z^n}{n^2}

, and

\displaystyle\sum_{n=1}^{\infty} \frac{z^n}{n}

. ((a)) Show that each series has radius of convergence 1. ((b)) Show that the first series converges nowhere on

C_1(0) = \{z : |z| = 1\}

. ((c)) Show that the second series converges everywhere on

C_1(0)

. ((d)) It turns out that the third series converges everywhere on

C_1(0)

, except at the point

z = 1

. This is not easy to prove. Give it a try.