Category:Definitions/Divergent Products

This category contains definitions related to Divergent Products.
Related results can be found in Category:Divergent Products.

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:

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