This week in class, we covered one way to sum an arithmetico-geometric series. Now we're going to cover a different approach.

Question

Let |r| < 1,
$$S = \sum_{k=0}^{\infty} r^k,$$and
$$T = \sum_{k=0}^{\infty} k r^k.$$Our approach is to write $T$ as a geometric series in terms of $S$ and $r$.

Give a closed form expression for $T$ in terms of $r$.