Definition:Continued Product/Inequality
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Definition
Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
The continued product of $\tuple {a_1, a_2, \ldots, a_n}$ can be written:
- $\ds \prod_{1 \mathop \le j \mathop \le n} a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$
Multiplicand
The set of elements $\set {a_j \in S}$ is called the multiplicand.
Notation
The sign $\ds \prod$ is called the product sign and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of product.
Also see
- Results about continued products can be found here.
Historical Note
The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.