Complex series that diverges at three points of $|z|=1$

Question

Does there exist a complex series that the radius of convergence is $R=1$ but the series diverges at exactly three points of a set $\{z \in \Bbb{C} : \space |z|=1 \}?$

I found this exercise and I wonder how should it be solved. I know that in real analysis there exist such series that the radius of convergence is $1$ in the interval $X$ but the same series diverges at the endpoints of $X$. But it has only two points of divergence.

Can anyone show me the way?

Answer 1

$$\sum_{n=1}^\infty\frac{z^n}n$$ converges at all points of the unit circle except $z=1$. So $$\sum_{n=1}^\infty\frac{z^{2n}}n$$ converges at all points of the unit circle except $z=1$ and $z=-1$. Can you now get a series that converges on the unit circle save for three points?