Definition:Absolutely Convergent Series/Real Numbers

Definition

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 $a_n$.

Also see

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

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.20$
• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Definition $1.2$