Complex Analysis: Theory and Applications · Problem 7.2.5

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · taylor-series

Problem

Show that

f(z) = \dfrac{1}{1-z}

has its Taylor series representation about the point

\alpha = i

given by

f(z) = \sum_{n=0}^{\infty}\frac{(z-i)^n}{(1-i)^{n+1}}, \quad\text{for all } z \in D_{\sqrt{2}}(i).