Sum of Infinite Geometric Sequence/Proof 2

Theorem
Let $S$ be a standard number field, i.e. $\Q$, $\R$ or $\C$.

Let $z \in S$.

Let $\left \vert {z}\right \vert < 1$, where $\left \vert {z}\right \vert$ denotes:
 * the absolute value of $z$, for real and rational $z$
 * the complex modulus of $z$ for complex $z$.

Then $\displaystyle \sum_{n \mathop = 0}^\infty z^n$ converges absolutely to $\dfrac 1 {1 - z}$.

Proof
By the Chain Rule and the corollary to Nth Derivative of Reciprocal of Mth 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.