Absolutely Convergent Series is Convergent/Real Numbers
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Theorem
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $\R$.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Proof
By the definition of absolute convergence, we have that $\ds \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent.
From the Comparison Test we have that $\ds \sum_{n \mathop = 1}^\infty a_n$ converges if and only if:
- $\forall n \in \N_{>0}: \size {a_n} \le b_n$
where $\ds \sum_{n \mathop = 1}^\infty b_n$ is convergent.
So substituting $\size {a_n}$ for $b_n$ in the above, the result follows.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.21$