University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.2 - Infinite Series - Exercises - Page 498: 39


Converges to $\dfrac{-\pi}{6}$

Work Step by Step

Since, we have $f(x)=\cos^{-1} x$ Here, $ s_n=[f(\dfrac{1}{2}-f(\dfrac{1}{3}]+[f(\dfrac{1}{3}-f(\dfrac{1}{4}]......[f(\dfrac{1}{k+1}-f(\dfrac{1}{k+2}]$ and $s_n=\cos^{-1}{\dfrac{1}{2}}-\cos ^{-1} \dfrac{1}{k+2}$ Thus, we have $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} [\cos^{-1}{\dfrac{1}{2}}-\cos ^{-1} \dfrac{1}{k+2}]=[\dfrac{\pi}{3}-\dfrac{\pi}{2}]=\dfrac{-\pi}{6}$ Thus, the given series converges to $\dfrac{-\pi}{6}$
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