Definition:Series/General

Definition

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

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

Informally, a series is what results when an infinite product is taken of $\sequence {a_n}$:

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

Formally, a series is a sequence in $S$.

Sequence of Partial Products

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 $\ds \sum_{n \mathop = 1}^\infty a_n$.