Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.5 - Alternating Series - 11.5 Exercises - Page 736: 32

Answer

$p\gt 0$

Work Step by Step

Case 1: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p\lt 0$ Now, $\lim\limits_{n \to \infty}(-1)^{n-1} n^{|p|}=DNE$ So, the series will not converge by the Test of Divergence. Case 2: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p = 0$ Now, $\lim\limits_{n \to \infty}(-1)^{n-1} n^{0}=DNE$ . So, the series will not converge by the Test of Divergence. Case 3: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p\gt 0$ Now, $\lim\limits_{n \to \infty}\dfrac{1}{n^p}=0$ . Here, the limit $0$ satisfies all the conditions for the alternating series test and so, the given series will converge for the value of $p\gt 0$ by the Test of Divergence.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.