# Definition:Absolute Convergence of Product/Complex Numbers

## Definition

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

### Definition 1

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.

### Definition 2

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

### Definition 3

The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent if and only if there exists $n_0 \in \N$ such that:

$a_n \ne -1$ for $n > n_0$
The series $\ds \sum_{n \mathop = n_0 + 1}^\infty \log \paren {1 + a_n}$ is absolutely convergent

where $\log$ denotes the complex logarithm.