Definition:Absolute Convergence of Product/Complex Numbers
Jump to navigation
Jump to search
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.