TITLE: Power series that diverges in specified number of points on the circle of convergence
QUESTION [6 upvotes]: I want to construct a complex power series with the radius of convergence $R=1$ that diverges in:
1) $k$ points on the circle $\{|z|=1\}$.
2) countable number of points on the circle
I have already crafted series that converges on the whole circle and diverges on the whole circle.
Can anybody help with the problem?
Update: I have googled a series $\sum{\frac{z^{kn}}{kn}}$ that is an answer for 1). Though I still need help for the countable case.

REPLY [2 votes]: Hint: for (2), take a sequence of points $z_k$ on the circle converging to $1$, say, and try
putting together series that diverge at each $z_k$.