Sum of Infinite Geometric Sequence

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 $\dfrac 1 {1 - z}$.

Corollary 1
With the same restriction on $z \in S$:

Corollary 2
With the same restriction on $z \in S$: