Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.3 The Integral Test and Estimates of Sums - 11.3 Exercises - Page 766: 26



Work Step by Step

$\int_{1}^{\infty}\frac{x}{x^{4}+1}=\lim\limits_{t \to \infty}\int_{1}^{\infty} \frac{x}{x^{4}+1}=\lim\limits_{t \to \infty}[\frac{1}{2}tan^{-1}(x^{2})]^{t}_{1}=\frac{1}{2}[tan^{-1}(\infty)-tan^{-1}(1)]=\frac{1}{2}[\frac{\pi}{2}-\frac{\pi}{4}]$ $=\frac{\pi}{8}$ Hence, the given series is convergent.
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