Power series that diverges in specified number of points on the circle of convergence

For part 1), we can use Abel's test to prove $\log(1-z)$ converges everywhere on the boundary except at $z=1$. I believe now taking $f(z) = \log(1-z^k)$ should do.

For part 2)

The generating function for the number of partitions of $n$ has a countable number of singularities and to deal with that, Hardy, Ramanujan and Littlewood came up with the Circle Method, which they used to determine the asymptotic behaviour of the partition number.

But, I suppose there is an easier example, as this is homework.

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 $z_k$.