Convergent series

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 $\sum_{n=1}^\infty a_n = S$.

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

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 $\frac{1}{1-r}$).