Definition:Convergent Series/Number Field

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

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

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

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 $s$, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

A series is said to be convergent it converges to some $s$.