Category:Definitions/Infinite Products

This category contains definitions related to Infinite Products.
Related results can be found in Category:Infinite Products.

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.