Sum of Infinite Geometric Sequence/Proof 2

Proof
By the Chain Rule and the corollary to $n$th Derivative of Reciprocal of $m$th Power:
 * $\dfrac {\mathrm d^n}{\mathrm dz^n} \dfrac 1 {1 - z} = \dfrac {n!} {\left({1 - z}\right)^{n + 1}}$

Thus the Maclaurin series expansion of $\dfrac 1 {1 - z}$ is:
 * $\displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!} \dfrac {n!} {\left({1 - 0}\right)^{n + 1}}$

whence the result.