# 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

- ,

then

converges.

**Proof:**