Definition:Convergent Series

Definition
Let $S$ be one of the standard number fields $\Q, \R, \C$.

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

Let $\left \langle {s_N} \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n=1}^\infty a_n$.

It follows that $\left \langle {s_N} \right \rangle$ can be treated as a sequence in the metric space $S$.

If $s_N \to s$ as $n \to \infty$, the series converges to the sum $s$.

Divergent Series
A series which is not convergent is divergent.