# Definition:Absolutely Convergent Series

## Definition

### General Definition

Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$.

Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ in $V$ is absolutely convergent if and only if $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ is a convergent series in $\R$.

### Real Numbers

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

Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent if and only if:

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

where $\size {a_n}$ denotes the absolute value of $a_n$.

### Complex Numbers

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 known as

An absolutely convergent series is also referred to as absolutely summable.

## Also see

• Results about absolute convergence can be found here.