Definition:Absolute Convergence of Product/General Definition/Definition 1

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Definition

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.

Let $\sequence{a_n}$ be a sequence in $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + a_n}$ is absolutely convergent if and only if $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + \norm{a_n}}$ is convergent.


Also see