# Having trouble with this infinite series and deciding whether it

Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
$$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{n}{\left({\frac{{{1}}}{{{2}{i}}}}\right)}^{{n}}$$

• Questions are typically answered in as fast as 30 minutes

### Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Bubich13
First let’s look if the series converges absolutely.
For this, we need to see if $$\displaystyle\sum{b}_{{n}}=\sum{\frac{{{n}}}{{{2}^{{n}}}}}$$ converges. And this is immediate using the ratio test
as $$\displaystyle\lim_{{{n}\to\infty}}{\frac{{{b}_{{{n}+{1}}}}}{{{b}_{{n}}}}}=\frac{{1}}{{2}}{ < }{1}$$
Conclusion: the given series converges absolutely hence converges
###### Not exactly what you’re looking for?
Chanell Sanborn
Hint. Your first thought is correct: look at the modulus.
Your reasoning about $$\displaystyle\infty\cdot{0}$$ is wrong.
Try the ratio test.
If you know about the geometric series
$$\displaystyle{1}+{x}+{x}^{{2}}+\ldots$$
you can differentiate, multiply by x and actually find out what your series converges to.
karton

Consider $$\sum_{n=0}^\infty nz^n$$. The radius of convergence is $$r=\lim_{n\to\infty}\frac{1}{n^{\frac{1}{n}}}=1$$. Since $$|\frac{1}{2i}|=\frac{1}{2}$$, the series converges.