# Sum of Infinite Geometric Sequence/Proof 4

## Theorem

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

## Proof

 $\ds \frac 1 {1 - z}$ $=$ $\ds \frac 1 {1 + \paren {-z} }$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-z}^k$ Power Series Expansion of $\dfrac 1 {1 + z}$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {-1}^k z^k$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^\infty \paren {-1}^{2 k} z^k$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^\infty z^k$

$\blacksquare$