Definition:Divergent Product/Divergence to Zero
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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 $\ds \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$
the product diverges to $0$, and we assign the value:
- $\ds \prod_{n \mathop = 1}^\infty a_n = 0$