# Definition:Absolutely Convergent Series

## Contents

## Definition

### General Definition

Let $V$ be a normed vector space with norm $\left\Vert{\cdot}\right\Vert$.

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

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ in $V$ is **absolutely convergent** if and only if $\displaystyle \sum_{n \mathop = 1}^\infty \left\Vert{a_n}\right\Vert$ is a convergent series in $\R$.

### Real Numbers

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

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

- $\displaystyle \sum_{n \mathop = 1}^\infty \left|{a_n}\right|$ is convergent

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

### Complex Numbers

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

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

- $\displaystyle \sum_{n \mathop = 1}^\infty \cmod {a_n}$ 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

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

- Absolutely Convergent Series is Convergent, which shows that an
**absolutely convergent series**in a Banach space is also convergent.

- Results about
**absolute convergence**can be found here.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**absolutely convergent series, absolutely summable**