calculus - Formula for $r+2r^2+3r^3+...+nr^n$

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I cannot think of a single one. Can you please help me with this problem?

Answer

We have
$$1+r+r^2 + \cdots + r^n = \dfrac{r^{n+1}-1}{r-1}$$
Differentiating this once, we obtain
$$1+2r + 3r^2 + \cdots + nr^{n-1}= \dfrac{nr^{n+1}-(n+1)r^n + 1}{(r-1)^2}$$

Multiply the above by $r$ to obtain what you want.