Geometric series

The idea

A geometric sequence multiplies by the same ratio $r$. The $n$th term is $ar^{n-1}$, the sum of $n$ terms is $S_n = \frac{a(1 - r^n)}{1 - r}$, and if $|r| < 1$ the sum to infinity is $S_\infty = \frac{a}{1 - r}$.

1. Find $r$ by dividing a term by the one before.
2. Use $ar^{n-1}$ for a term and $\frac{a(1 - r^n)}{1 - r}$ for a sum.
3. Use $S_\infty$ only when $-1 < r < 1$.

Watch out A sum to infinity needs $|r| < 1$, not just $r < 1$.