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.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. 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