Complex Analysis: Theory and Applications · Problem 7.3.19

Taylor and Laurent Series — Howell & Matthews

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

Problem

Use the Weierstrass

M

-test to show that the series

\displaystyle\sum_{n=1}^{\infty} c_{-n}(z-\alpha)^{-n}

of Theorem 7.7 converges uniformly on the set

\{z : |z-\alpha| \geq s\}

as claimed.