# Sum of Infinite Geometric Sequence/Corollary 1

## Corollary to Sum of Infinite Geometric Sequence

Let $S$ be a standard number field, i.e. $\Q$, $\R$ 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:

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

## Proof 1

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

$\blacksquare$

## Proof 2

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

$\blacksquare$