Sum of Infinite Geometric Sequence/Corollary 1/Proof 1
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Corollary to Sum of Infinite Geometric Sequence
- $\ds \sum_{n \mathop = 1}^\infty z^n = \frac z {1 - z}$
Proof
| \(\ds \sum_{n \mathop = 1}^\infty z^n\) | \(=\) | \(\ds -z^0 + \sum_{n \mathop = 0}^\infty z^n\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1 + \frac 1 {1 - z}\) | Sum of Infinite Geometric Sequence | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {z - 1 + 1} {1 - z}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac z {1 - z}\) |
$\blacksquare$