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?

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
$$\sum\limits_{n = 1}^{\infty} nr^{n - 1} = \frac{1}{(1 - r)^2}$$
This can be adapted for your series:
\begin{align}
\sum\limits_{x = 3}^{\infty} x \left(\frac 5 8 \right)^x &= \frac 5 8 \sum\limits_{x = 3}^{\infty} x \left(\frac 5 8\right)^{x - 1} \\
&= \frac 5 8 \left(\sum\limits_{x = 1}^{\infty} x \left(\frac 5 8 \right)^{x - 1} - 1 - 2 \cdot \frac 5 8\right) \\
&= \frac 5 8 \left(\frac{1}{(1 - (5/8))^2}-1 - \frac {10} 8\right)
\end{align}
