3
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.