University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Section 9.9 - Convergence of Taylor Series - Exercises - Page 542: 35


Error $\leq 4.2 \times 10^{-6}$

Work Step by Step

The Taylor series for $\sin x $ can be defined as: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ We have $ f(x)=\sin x \\ f^{,}(x) =\cos x \\ f^{,}(x) =-\sin x\\ ......\\ f^{4}(x) =\sin x $ Need to find $|f^{4} | \leq M $. So, $|R_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!}$ and $|R_3(0.1)| \leq (1) \times \dfrac{|0.1-0|^{4}}{4!} \approx 4.2 \times 10^{-6}$ So, Error $\leq 4.2 \times 10^{-6}$
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