# infinite summation formula help

How do I find the following sum? $$\sum_{x=3}^\infty 1.536x\left(\frac{5}{8}\right)^x$$ It wouldn't be geometric because of the $x$ in front, right?

• – Jonas Meyer Oct 14 '13 at 1:21

Hint: You are correct that it's not a geometric series. What we can do is note that

$$\sum\limits_{n = 0}^{\infty} r^n = \frac{1}{1 - r}$$

Now differentiating termwise, we see that

$$\sum\limits_{n = 1}^{\infty} nr^{n - 1} = \frac{1}{(1 - r)^2}$$