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: 35

Answer

Absolutely Convergent

Work Step by Step

A series $ \Sigma a_n$ is said to be absolutely convergent when $ \Sigma |a_n|$ is convergent. We notice that $ \Sigma_{n=1}^{\infty} |\dfrac{\cos ( n \pi) }{n \sqrt n}|= \Sigma_{n=1}^{\infty} \dfrac{1}{n^{3/2}}$ We see a p-series with common ratio $r=1.5 \gt 1$; when the common ratio $r \gt 1$, then a p-series is convergent. Therefore, the given series is Absolutely Convergent.
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