Convergent series - Imported Knowlpedia Content

A convergent series is a series $\sum_{n=1}^\infty a_n$ whose partial sums $s_n=\sum_{k=1}^n a_k$ converge to a finite limit $S$ , in which case one writes $\sum_{n=1}^\infty a_n = S$ .

Examples

$\sum_{n=0}^\infty \frac{1}{2^n}$ converges (in fact, to $2$ ).
For $|r|<1$ , the geometric series $\sum_{n=0}^\infty r^n$ converges (to $\frac{1}{1-r}$ ).

Equivalent characterizations

Equivalently, the partial sums form a convergent sequence in $\mathbb{R}$ . Two important refinements are absolute convergence and conditional convergence.