Give an example, with justification, of a power series $\sum_{n=0}^\infty a_nz^n$ such that the series converges for all $z$ such that $|z|=2$, but diverges if $|z|>2$.
I've been thinking about this for a week now, and all I can think of is that this is basically asking to come up with an example of a power series which is convergent on its radius of convergence ($R=2$). But I'm not sure how to construct something with a convergent radius. I don't think we really talked about it in my class, and I haven't found anything to clearly explain it yet. Thanks.
On the boundary of the circle of convergence you have $|a_n z^n| = |a_n| 2^ n$, so one method to make the power series converge on the boundary is to choose the $a_n$ such that $\sum_{n=0}^\infty |a_n| 2^n < \infty$. A simple choice would be $a_n 2^n = \frac{1}{(n+1)^2}$, i.e. the power series $$ \sum_{n=0}^\infty \frac{z^n}{2^n (n+1)^2} $$ This converges for $|z| \le 2$, but not for $|z| > 2$.