# Definition:Continued Product

## 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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**continued product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**continued product** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**product notation**