\begin{definition}[Definition:Divergent Product/Divergence to Zero]
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\sequence {a_n}$ 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 $\ds \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$
the product '''diverges to $0$''', and we assign the value:
:$\ds \prod_{n \mathop = 1}^\infty a_n = 0$
Category:Definitions/Convergence
Category:Definitions/Infinite Products
\end{definition}