Definition:Convergent Series

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

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

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

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.