Geometric series

A geometric series is the sum of numbers that form a geometric progression with common ration $r$ and scale factor (or starting value) $α$ :

\sum_{n = 0}^{N - 1} α r^{n} = α r^{0} + α r^{1} + α r^{2} + \dots + α r^{N - 1} = \frac{α (1 - r^{N})}{1 - r} .

When the number $N$ of terms goes to infinity, the series converges if and only if $| r | < 1$ . In this case

\sum_{n = 0}^{\infty} α r^{n} = \frac{α}{1 - r}, | r | < 1 .

(A.20)