University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.6 - Alternating Series and Conditional Convergence - Exercises - Page 521: 48


Absolutely Convergent

Work Step by Step

A series $ \Sigma a_n$ is said to be absolutely convergent when $ \Sigma |a_n|$ is convergent. When the common ratio $r \gt 1$, then a p-series is convergent. To use the Limit Comparison Test, we consider a series $\Sigma a_n$ such that $A_n=(-1)^n B_n$; $B_n \geq 0$ for all $n$ It has been noticed that $B_n=\dfrac{1}{n^2}$ Now, $\lim\limits_{n \to \infty} \dfrac{A_n}{B_n}=\lim\limits_{n \to \infty} \dfrac{n^2}{(2n+1)^2}+ \dfrac{n^2}{(2n+2)^2}$ or, $=\lim\limits_{n \to \infty} \dfrac{1}{(2+1/n)^2}+ \dfrac{1}{(2+2/n)^2}$ or, $= \dfrac{1}{2} \ne 0 \ne \infty$ Therefore, the given series is Absolutely Convergent by the Limit Comparison Test.
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