Definition:Absolutely Convergent Series/Complex Numbers

Definition

Let $S = \ds \sum_{n \mathop = 1}^\infty a_n$ be a series in the complex number field $\C$.

Then $S$ is absolutely convergent if and only if:

$\ds \sum_{n \mathop = 1}^\infty \cmod {a_n}$ is convergent

where $\cmod {a_n}$ denotes the complex modulus of $a_n$.

Also see

• Definition:Conditionally Convergent Series, the antithesis of absolutely convergent series.

Examples

Example: $\paren {\dfrac z {1 - z} }^n$

The complex series defined as:

$\ds S = \sum_{n \mathop = 1}^\infty \paren {\dfrac z {1 - z} }^n$

is absolutely convergent, provided $\Re \paren z < \dfrac 1 2$.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series: $(4.9)$