Geometric Series

Sum of a finite geometric series: $$\sum_{k=0}^{n}{r^k}=1+r+r^2+\dots+r^n=\frac{1-r^{n+1}}{1-r}$$

Sum of an infinite geometric series (0 < r < 1): $$\sum_{k=0}^{\infty}{r^k}=1+r+r^2+\dots=\frac{1}{1-r}$$

Example: $$\begin{aligned} \sum_{k=2}^{\infty}{\frac{1}{2^k}}&=\frac{1}{4}+\frac{1}{8}+\dots \\ &=\frac{1}{4}\left(1+\frac{1}{2}+\frac{1}{4}+\dots\right) \\ &=\frac{1}{4}\left(\sum_{k=0}^{\infty}{\left(\frac{1}{2}\right)^k}\right) \\ &=\frac{1}{4}\cdot\frac{1}{1-\frac{1}{2}}=\frac{1}{2} \end{aligned}$$