A First Course in Complex Analysis · Problem 7.5.29

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

Mathematics — Complex Analysis · weierstrass-m-test-uniform-convergence

Problem

Use the Weierstra\ss\ MM-test to show that each of the following series converges uniformly on the given domain: (a) ∑k≥1zkk2\displaystyle\sum_{k\geq 1} \frac{z^k}{k^2} on D‾[0,1]\overline{D}[0,1] (b) ∑k≥01zk\displaystyle\sum_{k\geq 0} \frac{1}{z^k} on {z∈C:∣z∣≥2}\{z \in \mathbb{C} : |z| \geq 2\} (c) ∑k≥0zkzk+1\displaystyle\sum_{k\geq 0} \frac{z^k}{z^k+1} on D‾[0,r]\overline{D}[0,r] where 0≤r<10 \leq r < 1

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