Infinite Geometric Series

If $|r| < 1$, a geometric series converges even as $n \to \infty$. The sum $a + ar + ar^2 + ar^3 + ...$ equals $\frac{a}{1 - r}$.

For example, $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$ has $a = 1$ and $r = \frac{1}{2}$. The sum is $\frac{1}{1 - 1/2} = 2$.

This appears in probability (expected values), algorithm analysis (amortized cost), and anywhere you sum decreasing powers. When $|r| \geq 1$, the series diverges (sum grows without bound).