Definition:Convergent Product/Number Field/Nonzero Sequence

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Definition

Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$.

Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$.


Then:

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent

if and only if:

its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.