Definition:Absolute Convergence of Product

Definition

Complex Numbers

Let $\sequence {a_n}$ be a sequence in $\C$.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is absolutely convergent if and only if $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\vert{a_n}\right\vert}\right)$ is convergent.

General 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 presented as

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