# Definition:Continued Product/Infinite

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## Definition

### Indexed Infinite Product

Let $\struct {S, \times}$ be an algebraic structure.

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### Product over Set

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### Propositional Function

Let $\struct {S, \times}$ be an algebraic structure.

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.

## 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

## 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):**infinite product** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**product notation**