Consider the series of functions $\sum z^n/n$ defined over the complex numbers.
By the Cauchy-Hadamard radius of convergence we have convergence when $|z|<1$, and divergence when $|z|>1$.
Now, what happens when $|z|=1$? For the concrete cases $z=1$ and $z=-1$ I can figure out that it diverges and converges, respectively.
But I am stuck trying to analyze what happens when the imaginary part of $z$ is not zero.
Also, does the series uniformly converge in the open disk of radius 1? My guess is not because of the divergence in $z=1$. Is this argument correct?
