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) $α$ :

∑_{n=0}^{N−1} α r^{n} = α r^{0} + α r^{1} + α r^{2} + … + 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

∑_{n=0}^{∞} α r^{n} = \frac{α}{1 − r},|r| < 1.

(A.16)