Definition:Convergent Series

Definition
Let $\struct {S, \circ, \tau}$ be a topological semigroup.

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

This series is said to be convergent its sequence of partial sums $\sequence {s_N}$ converges in the topological space $\struct {S, \tau}$.

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

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

Also see

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