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 $R \left({j}\right)$.

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.