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?

First let’s look if the series converges absolutely.
For this, we need to see if $\sum b_n = \sum \frac{n}{2^n}$ converges. And this is immediate using the ratio test 
as $\lim\limits_{n\to \infty}\frac{b_{n+1}}{b_n} =1/2<1$.
Conclusion: the given series converges absolutely hence converges
