Definition:Absolute Convergence

General Definition

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

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

Then the series $\displaystyle \sum_{n=1}^\infty a_n$ in $V$ is said to be absolutely convergent iff $\displaystyle \sum_{n=1}^\infty \left\Vert{a_n}\right\Vert$ is a convergent series in $\R$.

An absolutely convergent series is also convergent, as proved on Absolutely Convergent Series is Convergent.

Definition in a Number Field

Let $\displaystyle \sum_{n=1}^\infty a_n$ be a series in one of the standard number fields $\Q, \R, \C$.

Then $\displaystyle \sum_{n=1}^\infty a_n$ is said to be absolutely convergent iff $\displaystyle \sum_{n=1}^\infty \left|{a_n}\right|$ is convergent.

Also see

• Conditional convergence, the opposite of absolute convergence.
• Absolutely Convergent Series is Convergent

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 6.20$