Definition:Absolutely Convergent Series/General
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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 $\R$.
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.