University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Section 9.10 - The Binomial Series and Applications of Taylor Series - Exercises - Page 549: 22


$\approx 0.4863853764$ with an error of about $1.9 \times 10^{-10}$

Work Step by Step

We integrate the integral with respect to $ x $ taking a lower limit as $0$ and upper limit as $1$. $\int_0^{1} \dfrac{1- \cos x}{x^2} dx=\int_0^{1} [0.5-\dfrac{x^2}{4 !}?+\dfrac{x^{4}}{6 !}-...] dx $ or, $=0.5-\dfrac{1}{3 \cdot 4!}+\dfrac{1}{5 \cdot 6!}-....$ But $\dfrac{1}{11 \cdot 12!} \approx 1.9 \times 10^{-10} $ and the proceeding term is greater than $10^{-8}$. So, the first five terms of the series would give the accuracy. $\int_0^{1} \dfrac{1- \cos x}{x^2} dx \approx 0.4863853764$ with an error of about $1.9 \times 10^{-10}$.
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