# Definition:Absolute Convergence of Product

## Definition

### Complex Numbers

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

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

## Also presented as

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