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: 45

Answer

Absolutely Convergent

Work Step by Step

We need to apply the Ratio Test to the series. $\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} \dfrac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$ or, $=\lim\limits_{n \to \infty} \dfrac{1+e^{-2n}}{e+e^{-2n-1}}$ or, $=\lim\limits_{n \to \infty} \dfrac{1+0}{e+0}$ So, $=\dfrac{1}{e} \lt 1$ Thus, the series is Absolutely Convergent by the ratio test.
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