Definition:Absolute Convergence of Product/Complex Numbers/Definition 3

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Definition

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


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ 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 $\displaystyle \sum_{n \mathop = n_0 + 1}^\infty \log \paren {1 + a_n}$ is absolutely convergent

where $\log$ denotes the complex logarithm.


Also see


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