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: 76



Work Step by Step

The Taylor series for $\cos x $ can be defined as: $\cos x=1-\dfrac{x^2}{2!}+\dfrac{ x^4}{4!}-....$ and the Taylor series for $\sin x $ can be defined as: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ Now, $\lim\limits_{h \to 0} (\dfrac{\sin h/h-\cos h}{h^2}=\lim\limits_{h \to 0} \dfrac{(1/h) (h-h^3/3!+h^5/5!-....)-(1-h^2/2!+h^4/4!-....) }{h^2}$ or, $=\lim\limits_{h \to 0} \dfrac{t^2-2+2 (1-\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....) }{2t^2 (1-1+\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....)}$ or, $=\lim\limits_{h \to 0} (\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{h^2}{5!}-\dfrac{h^2}{4!}+.....)$ or, $=\dfrac{1}{3}$
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