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?

What exactly is $r$? Real? Positive or Negative? Rational or Irrational? etc — Rhys Hughes, Jun 07 '18 at 18:10
Try to compute $(1-r)(r + 2r^2 + 3r^3+\ldots)$. It will give you a simple geometrical series — Winther, Jun 07 '18 at 18:11

score 2 · Accepted Answer · answered Jun 07 '18 at 18:10

2

HINT

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

answered Jun 07 '18 at 18:10

openspace

1 Answers1