Complex Analysis: Theory and Applications · Problem 7.1.4

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · uniform-convergence

Problem

Show that

S_n(z) = \displaystyle\sum_{k=0}^{n-1} z^k = \frac{1-z^n}{1-z}

does not converge uniformly to

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

on the set

T = D_1(0)

by appealing to Statement (7.3). Hint: Given

\varepsilon > 0

and a positive integer

n

, let

z_n = \varepsilon^{\frac{1}{n}}