# Definition:Divergent Product

## Definition

An infinite product which is not convergent is divergent.

### Divergence to zero

If either:

there exist infinitely many $n \in \N$ with $a_n = 0$
there exists $n_0 \in \N$ with $a_n \ne 0$ for all $n > n_0$ and the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$

the product diverges to $0$, and we assign the value:

$\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$

## Remark

A product may converge to $0$ as well.