# Definition:Divergent Product

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## Definition

An infinite product which is not convergent is **divergent**.

This article, or a section of it, needs explaining.In particular: Nelson separately defines an Definition:Oscillating Product which is one that is neither convergent nor divergent, but then does not rigorously define divergent. Research needed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.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 `{{Explain}}` from the code. |

### 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.

## Also see

- Results about
**divergent products**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**divergent product** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**infinite product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**divergent product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**infinite product**