# Definition:Product Notation (Algebra)/Infinite

## Contents

## Definition

### Indexed Infinite Product

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

### Product over Set

### 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 $\displaystyle \prod_{\map R j} a_j$ is:

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

provided that both limits exist.

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

## Multiplicand

The set of elements $\left\{{a_j \in S: 1 \le j \le n, R \left({j}\right)}\right\}$ is called the **multiplicand**.

## Notation

The sign $\displaystyle \prod$ 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

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