\begin{definition}[Definition:Convergent Series]
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''' {{iff}} 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 $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
\end{definition}