A First Course in Complex Analysis · Problem 7.5.30

Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · uniform-convergence-geometric-series

Problem

Fix

z \in \mathbb{C}

and

r > |z|

. Prove that

\displaystyle\sum_{k\geq 0} \left(\frac{z}{w}\right)^k

converges uniformly in the variable

w

for

|w| \geq r