Finding a complex power series which converges for specified bounds

Question

I've been asked to find an example of a complex power series which converges for certain z. I understand how to do this for an open ball, I just manipulated another power series to be centred on a given a with given radius of convergence R

eg, converges for every z in B(a, R) but diverges for any z not in B(a, R)

But I'm completely stuck on how to find a series which converges for every z in closed ball B(a, R) but diverges for any z not in the closed ball.

Any hints on where to start looking? How do I test that a given series converges absolutely on the boundary?

Answer 1

The series $\sum_{n=1}^\infty \frac{z^n}{n^2}$ converges everywhere in the closed unit disk, but diverges everywhere outside.

Answer 2

On $B(0,1)$, consider $$f(z)=\sum_{n=0}^\infty\frac{z^n}{n^2}$$ The radius of convergence is $1$ and it converges at every point with $|z|=1$. Modify this example for general $B(a,R).$