Definition:Divergent Product/Divergence to Zero

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

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