Definition:Product Notation (Algebra)/Infinite

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

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.

Also see

 * Definition:Convergent Product
 * Definition:Divergent Product