Complex Infinite Series Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
$$\sum_{n=1}^\infty n(\frac{1}{2i})^n$$
My thoughts are that you take the modulus of the fraction and get $\frac{1}{2}$ to the exponent $n$ makes it go to $0$ and then multiplied by $n$ make it 
$$\infty*0$$ which is always divergent right, making the series diverge? Can someone also clarify that this is the case?

Consider $\sum_{n=0}^\infty  nz^n$.  The radius of convergence is $r=\limsup_{n\to\infty}\frac1{n^{\frac1n}}=1$.  Since $\mid\frac1{2i}\mid=\frac12 $, the series converges. 
