# Sum of Infinite Geometric Sequence/Corollary 2

## Corollary to Sum of Infinite Geometric Sequence

Let $S$ be a standard number field, that is $\Q$, $\R$ or $\C$.

Let $z \in S$.

Let $\size z < 1$, where $\size z$ denotes:

the absolute value of $z$, for real and rational $z$
the complex modulus of $z$ for complex $z$.

Then:

$\displaystyle \sum_{n \mathop = 0}^\infty a z^n = \frac a {1 - z}$

## Proof

 $\ds \sum_{n \mathop = 0}^\infty a z^n$ $=$ $\ds a \sum_{n \mathop = 0}^\infty z^n$ $\ds$ $=$ $\ds a \frac 1 {1 - z}$ Sum of Infinite Geometric Sequence $\ds$ $=$ $\ds \frac a {1 - z}$

$\blacksquare$