I'm struggling with finding the sum of the series $\sum_{n=0}^{\infty}(n+1)(\frac{1}{4})^n [\math]<br />So far I've got:<br /><script type="math/tex">\sum_{n=0}^{\infty}(n+1)(\frac{1}{4})^n = \sum_{n=0}^{\infty}n(\frac{1}{4})^n + \sum_{n=0}^{\infty}(\frac{1}{4})^n = \sum_{n=0}^{\infty}n(\frac{1}{4})^n + \frac{4}{3}$
By a numerical approach I've found $\sum_{n=0}^{\infty}n(\frac{1}{4})^n \approx \frac{4}{9}$, but I have little clue on how to derive it. Any ideas?