Definition:Series/General/Sequence of Partial Products

Definition

Let $\struct {S, \circ}$ be a semigroup.

Let $\sequence {a_n}$ be a sequence in $S$.

Let $s$ be the the series:

$\ds s = \sum_{n \mathop = 1}^\infty a_n = a_1 \circ a_2 \circ a_3 \circ \cdots$

The sequence $\sequence {s_N}$ defined as the indexed iterated operation:

$\ds s_N = \sum_{n \mathop = 1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$

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