Definition:Absolutely Convergent Series

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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 $a_n$.


Examples

Arbitrary Example

Let $S$ be the series defined as:

\(\ds S\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \paren {\dfrac 1 n}^n\)
\(\ds \) \(=\) \(\ds 1 - \paren {\dfrac 1 2}^2 + \paren {\dfrac 1 3}^3 - \paren {\dfrac 1 4}^4 + \cdots\)

Then $S$ is absolutely convergent.


Also known as

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


Also see

  • Results about absolute convergence can be found here.


Sources

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