Geometric series

$\sum ar^{n}$ — converges to $\dfrac{a}{1-r}$ exactly when $|r|<1$ , diverges otherwise.

A geometric series $\sum_{n=0}^{\infty}ar^n$ has each term a fixed ratio $r$ times the previous one. It converges if and only if $|r|<1$ , in which case its sum is exactly $\dfrac{a}{1-r}$ — one of the few series types with a simple closed-form sum, derived from the finite partial-sum formula $S_N=a\dfrac{1-r^N}{1-r}$ and letting $N\to\infty$ .

Examples

$\sum_{n=0}^{\infty}3\left(\frac12\right)^n=\dfrac{3}{1-\frac12}=6$ (converges, since $|r|=\frac12<1$ ).
$\sum_{n=0}^{\infty}2\cdot3^n$ diverges, since $|r|=3\ge1$ .