# 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 $\ds \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$

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

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

## Remark

This page has been identified as a candidate for refactoring of advanced complexity.In particular: If a product may "converge to $0$", and it needs to be made explicit that it does (as is clear by the fact that it has been singled out for a specific "remark"), then it merits a separate page (possibly transcluded as required) and/or a specific "explanatory" page detailing the differences between all these types of convergence / divergence, as they may not be clear or obvious to all.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

A product may converge to $0$ as well.