Sequences and Series/Infinite products

Definition (infinite product):

Let be a sequence of numbers in or . If the limit

exists, it is called the infinite product of and denoted by


Proposition (necessary condition for convergence of infinite products):

In order for the infinite product

of a sequence to exist and not to be zero, it is necessary that


Proof: Suppose that not . Then there exists and an infinite sequence such that for all we have . Thus, upon denoting


we will have


Suppose for a contradiction that exited and was equal to . Then when is sufficiently large, we will have


which is a contradiction.

Proposition (series criterion for the convergence of infinite products):

Let be a sequence of real numbers. If