evaluating $\sum_{n=1}^{\infty}n\left(-\frac{1}{5}\right)^{n-1}$

Question

Evaluate with justification $$\sum_{n=1}^{\infty}n\left(-\frac{1}{5}\right)^{n-1}$$

I'm not quite sure how to evaluate this sum, any help would be appreciated.

Answer 1

Hint

Consider

$$\sum_{n=1}^\infty n x^{n-1}=\left(\sum_{n=1}^\infty x^{n}\right)'$$ When done, make $x=-\frac 15$.