# Geometric Series

## Formula

### Finite Geometric Series

• For $r \neq 1$
• $a + ar + ar^2 + ar^3 + ... + ar^{n - 1} = \displaystyle\sum_{k=0}^{n-1} ar^k = a(\frac{1 - r^n}{1 - r})$
• $a$ is the first term
• $r$ is the common ratio
• $n$ is the term you are looking for

## Infinite Geometric Series

• Absolute value for $r$ must be less than $1$ for the series to converge
• If $r \gt 1$ then the sum is $\infty$
• $a + ar + ar^2 + ar^3 + ... + ar^{n - 1} = \displaystyle\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}$