Definition:Divergent Product/Divergence to Zero

Definition
Let $\struct {\mathbb K, \size{\,\cdot\,}}$ be a valued field.

Let $\left\langle{a_n}\right\rangle$ be a sequence of elements of $\mathbb K$.

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

the product diverges to $0$, and we assign the value $\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$.