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 - Practice Exercises - Page 553: 75



Work Step by Step

The Taylor series for $\cos x $ can be defined as: $\cos x=1-\dfrac{x^2}{2!}+\dfrac{ x^4}{4!}-....$ Now, $\lim\limits_{t \to 0} (\dfrac{1}{2-2 \cos t}-\dfrac{1}{t^2}=\lim\limits_{t \to 0} \dfrac{t^2-2+2 \cos t }{2t^2 (1-\cos t)}$ or, $=\lim\limits_{t \to 0} \dfrac{t^2-2+2 (1-\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....) }{2t^2 (1-1+\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....)}$ or, $=\lim\limits_{t \to 0} \dfrac{2 (\dfrac{1}{4!}-\dfrac{ t^2}{6!}-....) }({1-\dfrac{2t^2}{4!}+...})$ or, $=\dfrac{1}{12}$
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