Definition:Series/General/Sequence of Partial Products

Definition
Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left \langle {a_n} \right \rangle$ be a sequence in $S$.

Let $s$ be the the series:
 * $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n = a_1 \circ a_2 \circ a_3 \circ \cdots$

The sequence $\left \langle {s_N} \right \rangle$ defined as the indexed iterated operation:
 * $\displaystyle s_N = \sum_{n \mathop = 1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$

is the sequence of partial products of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.