Consider a power series $$ \sum_{n=0}^{\infty}a_nz^n $$ where $a_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that for $|z|<R$ the series converges absolutely; for $|z|>R$ it does not converge.

On the other hand, when $|z|=R$, the series can have very different behaviors. This has been discussed in many posts, e.g. Behaviour of power series on their circle of convergence , Seeking a Geometric Proof of a Generalized Alternating Series' Convergence .

I am looking for some relatively easy explicit examples of power series with funny behavior at the boundary. The only I know is $$ \sum_n\frac{1}{n}z^n $$ or small variations, e.g. replacing $z$ with $z^k$.

thanks

wouldconverge every place else. Suppose $\sum c_k$ diverges with the $c_i$ positive and decreasing to $0.$ Then $\sum z^{2^k}c_k$ would diverge to infinity at a $2^j$th root of unity but should converge everywhere else. $\endgroup$ – Aaron Meyerowitz Oct 3 '14 at 8:48