Geometric series

Infinite geometric series

$$\sum _{n=0}^{\infty }a{r}^n=\sum _{n=1}^{\infty }a{r}^{n-1}=\frac{a}{1-r}$$

for $|r|<1$

Finite geometric series

$$\sum _{k=0}^{n-1}a{r}^k=\sum _{k=1}^{n}a{r}^{k-1}=\{\begin{array}{l}a\frac{1-r^n}{1-r},r\ne 1\\ an,r=1\end{array}$$