Show that $\sum_{k = 0}^{\infty} k^2x^k = x(1 + x)/(1 - x)^3$ for $% $.

From chapter text $\sum_{k = 0}^{\infty} kx^k = \frac x {(1 - x)^2}$ (Eqn. A.8).

By differentiating this with respect to $x$, we get:

Multiplying both sides by $x$, we get: