Definition:Product Notation (Algebra)/Infinite

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Indexed Infinite Product

Let $\left({S, \times}\right)$ be an algebraic structure.

Product over Set

Propositional Function

Let $\left({S, \times}\right)$ be an algebraic structure.

Let an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$.

Then the precise meaning of $\displaystyle \prod_{R \left({j}\right)} a_j$ is:

$\displaystyle \prod_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0} } a_j}\right) \times \left({\lim_{n \mathop \to \infty} \prod_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$

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.


Historical Note

The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.


Also see