# Definition:Continued Product/Infinite

This page has been identified as a candidate for refactoring of basic complexity.In particular: First complete the defs, then make subpagesUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## Definition

### Indexed Infinite Product

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

This definition needs to be completed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### Product over Set

This definition needs to be completed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### 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

- Results about
**infinite 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** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**infinite product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**continued product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**infinite 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**