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: 31



Work Step by Step

Taylor series for $\cos t $ can be defined as: $\cos t=1-\dfrac{t^2}{2}+\dfrac{t^4}{4}-\dfrac{t^6}{6}+....$ Now, $\lim\limits_{t \to 0}\dfrac{1-\cos t-\dfrac{t^2}{2}}{t^4}=\lim\limits_{t \to 0} \dfrac{(1-\dfrac{t^2}{2})-(1-\dfrac{t^2}{2}+\dfrac{t^4}{4}-\dfrac{t^6}{6}+....)}{t^4}$ or, $=\lim\limits_{t \to 0} \dfrac{-\dfrac{t^4}{4!}+\dfrac{t^6}{6!}+...}{t^4}$ or, $=\lim\limits_{x \to 0} [-\dfrac{1}{4!}+\dfrac{t^2}{6!}+....]$ or, $=-\dfrac{1}{24}$
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