Definition:Continued Product

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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 $S$.


Definition by Index

The composite is called the continued product of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

$\ds \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$


Definition by Inequality

The continued product of $\tuple {a_1, a_2, \ldots, a_n}$ can be written:

$\ds \prod_{1 \mathop \le j \mathop \le n} a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$


Definition by Propositional Function

Let $\map R j$ be a propositional function of $j$.

Then we can write:

$\ds \prod_{\map R j} a_j = \text { the product of all $a_j$ such that $\map R j$ holds}$.


If more than one propositional function is written under the product sign, they must all hold.


Infinite

Let an infinite number of values of $j$ satisfy the propositional function $\map R j$.

Then the precise meaning of $\ds \prod_{\map R j} a_j$ is:

$\ds \prod_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ -n \mathop \le j \mathop < 0} } a_j} \times \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$

provided that both limits exist.

If either limit does fail to exist, then the infinite product does not exist.


Index Variable

Consider the continued product, in either of the three forms:

$\ds \prod_{j \mathop = 1}^n a_j \qquad \prod_{1 \mathop \le j \mathop \le n} a_j \qquad \prod_{\map R j} a_j$


The variable $j$, an example of a bound variable, is known as the index variable of the continued product.


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.


Vacuous Product

Take the composite expressed as a continued product:

$\ds \prod_{\map R j} a_j$

where $\map R j$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\map R j$ is true.

Then $\ds \prod_{\map R j} a_j$ is defined to be $1$.

Beware: not zero.

This composite is called a vacuous product.


This is because:

$\forall a: a \times 1 = a$

where $a$ is a number.

Hence for all $j$ for which $\map R j$ is false, the value of the product is unaffected.


This is most frequently seen in the form:

$\ds \prod_{j \mathop = m}^n a_j = 1$

where $m > n$.

In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$.


Also known as

A continued product can also be seen as product notation, but such a term is not only imprecise but also ambiguous.


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.


Sources

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