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

Definition

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 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.

Also see

• Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers

Sources

• 1973: John B. Conway: Functions of One Complex Variable $\text {VII}$: Compact and Convergence in the Space of Analytic Functions: $\S 5$: Weierstrass Factorization Theorem: Definition $5.5$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely convergent: 2. (of an infinite product)