I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for $|z|<1$: $$ \sum_{n=1}^{\infty} \frac{z^n}{n^2} $$ I'm not really sure how to proceed with this question. I know that the Weierstrass M test would give me uniform convergence if I could find a series of real numbers that are always larger than the magnitude of the terms in my complex series, but I the only series I can think of is $\sum_{n=1}^{\infty} \frac{1}{n} > \sum_{n=1}^{\infty} |\frac{z^n}{n^2}|$ and that diverges.

Any suggestions would be greatly appreciated.


The OP has the correct idea of using the Weierstrass M-test. The series that would work for this question is $\frac{1}{n^2}$.

Let $M_n = \frac{1}{n^2}$, so

$ \forall n\geq 1, \forall z \in \mathbb{C} \ s.t \ |z| < 1$

\begin{equation} \frac{|z|}{n^2} \leq M_n \end{equation}

and the series $ \sum_{n=1}^{\infty} M_n$ converges, therefore by Weierstrass M-Test the given series $\sum_{n=1}^\infty \frac{z}{n^2}$ coverages uniformly.

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