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

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 a and r .

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