Squeeze Theorem for Absolutely Convergent Series

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Theorem

Let $\ds \sum \size {a_n}$ be an absolutely convergent series in $\R$.

Suppose that:

$\ds -\sum \size {a_n} = \sum \size {a_n}$


Then $\ds \sum a_n$ equals the above two series.


Proof

From Absolutely Convergent Real Series is Convergent, the convergence of:

$\ds \sum_{n \mathop = 1}^\infty \size {a_n}$

implies that of:

$\ds \sum_{n \mathop = 1}^\infty a_n$


By Negative of Absolute Value:

$\ds -\size {\sum_{n \mathop = 1}^j a_n} \le \sum_{n \mathop = 1}^j a_n \le \size {\sum_{n \mathop = 1}^j a_n}$


By repeated application of Triangle Inequality:

$\ds -\sum_{n \mathop = 1}^j \size {a_n} \le \sum_{n \mathop = 1}^j a_n \le \sum_{n \mathop = 1}^j \size {a_n}$


We have by hypothesis that the leftmost and rightmost terms converge as $j \to +\infty$.

Hence the result, from Squeeze Theorem for Real Sequences.

$\blacksquare$