Definition:Divergent Product/Divergence to Zero

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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:

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