Let , \(|r| < 1\) ,
\(S = \sum^{\infty}_{k=0}r^k\)
and
\(T = \sum^{\infty}_{k=0}kr^k\)
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
Let f(r) = 1 + r + r^2 + ... = 1/(1 - r).
Taking the derivative: f'(r) = 1 + 2r + 3r^2 + ... = 2r/(1 - r)^2.
Then r + 2r^2 + 3r^3 + ... = 2r^2/(1 - r)^2.
Therefore, T = 2r^2/(1 - r)^2.
Like so:
