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: 33



Work Step by Step

Taylor series for $\tan^{-1} y $ can be defined as: $\tan^{-1} y= y-\dfrac{y^3}{3}+\dfrac{y^5}{5}-....$ Now, $\lim\limits_{y \to 0}\dfrac{y-\tan^{-1} y}{y^3}=\lim\limits_{y \to 0} \dfrac{[y-(y-\dfrac{y^3}{3}+\dfrac{y^5}{5}-....)]}{y^3}$ or, $=\lim\limits_{y \to 0} \dfrac{\dfrac{y^3}{3}- \dfrac{y^5}{5}+...}{y^3}$ or, $=\lim\limits_{y \to 0} [\dfrac{1}{3}-\dfrac{y^2}{5}+\dfrac{y^4}{7}-....]$ or, $=\dfrac{1}{3}$
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