# Definition:Divergent Product

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## 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.