Definition:Convergent Series

Definition
Let $\left({S, \circ, \tau}\right)$ be a topological semigroup.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

This series is said to be convergent its sequence of partial sums $\left \langle {s_N} \right \rangle$ converges in the topological space $\left({S, \tau}\right)$.

If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

Also known as
A convergent series is also known as a summable series.

Also see

 * Definition:Absolutely Convergent Series
 * Definition:Conditionally Convergent Series