How does one evaluate $\sum_{t=1}^\infty tr^t$?

Question

WolframAlpha says that $\sum_{t=1}^\infty tr^t=\frac{r}{(r-1)^2}$, which seems reasonable. But how can one derive this by hand?

Also, is there a name for this series?

Answer 1

HINT

consider $\sum_{t \ge 1} r^{t}$. We can differentiate it (why?). When consider it first derivate and compare with your object.