Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 37


Absolutely Convergent

Work Step by Step

Let us apply the Root Test to the series. $L=\lim\limits_{n \to \infty} |a_n|^{1/n}=\lim\limits_{n \to \infty} |\dfrac{(-1)^n(n+1)^n}{(2n)^n}| \\ =\Sigma_{n=1}^{\infty} \dfrac{n+1}{2n} \\=\Sigma_{n=1}^{\infty} \dfrac{1+1/n}{\dfrac{2n}{n}} \\=\dfrac{1}{2} \lt 1$ This implies that the series is Absolutely Convergent by the root test.
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