Definition:Series/General
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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 $\ds \sum_{n \mathop = 1}^\infty a_n$.