Definition:Absolutely Convergent Series

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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 $\left|{a_n}\right|$ 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 see

  • Results about absolute convergence can be found here.