About complex power series I have a really big doubt. I'm trying to find all the values of $z$ for which the next power series converges: $$\sum_{n=0}^{\infty} \frac{z^{3n}}{8^{n}(1-in)} $$
Using the root test I have that $$\lim_{n\to \infty}\sqrt[n]{\left|\frac{z^{3n}}{8^{n}(1-in)}\right|} = \lim_{n\to\infty}\left|\frac{z^3}{8(1-in)^{1/n}} \right| = \left|\frac{z^3}{8}\right|<1 \Leftrightarrow |z|<2$$
So, here I haven't any problem. My problem is when I try to study the edge $|z|=2$. I been trying to use the Dirichlet's test, but the convergence of the serie depends of $\arg(z)$. If anyone can help me, really I appreciate it.

HINT:
The series $\sum_{n=1}^\infty \frac{e^{inx}}{n}$ converges whenever $x\ne 2\ell \pi$ for integer-values of $\ell$.  
Here, the series of interest is asymptotically $i\sum_{n=1}^\infty \frac{e^{inx}}{n}$ with $x=3\arg(z)$.  
