Definition:Divergent Product

Definition
An infinite product which is not convergent is divergent.

If either:
 * There exist infinitely many $n\in\N$ with $a_n=0$
 * There exists $n_0\in\N$ with $a_n\neq0$ 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 $\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$.

Also see

 * Definition:Divergent Series
 * Definition:Convergent Product