Sum of Infinite Geometric Sequence/Corollary 1
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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:
- $\ds \sum_{n \mathop = 1}^\infty z^n = \frac z {1 - z}$
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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac z {1 - z}\) |
$\blacksquare$