Squeeze Theorem for Absolutely Convergent Series
Jump to navigation
Jump to search
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.
$\blacksquare$