\begin{definition}[Definition:Convergent Series/Number Field]
Let $S$ be one of the standard number fields $\Q, \R, \C$.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
Let $\sequence {s_N}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
It follows that $\sequence {s_N}$ 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 $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
A series is said to be '''convergent''' {{iff}} it converges to some $s$.
\end{definition}