Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.9 - Convergence of Taylor Series - Exercises 10.9 - Page 625: 35

Answer

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

Work Step by Step

Recall the Taylor series for $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ $ f(x)=\sin x \\ f^{,}(x) =\cos x \\ f^{,}(x) =-\sin x\\ ......\\ f^{4}(x) =\sin x $ We will compute $|f^{4} | \leq M $. $|R_n(x)| \leq M \dfrac{|x-a|^{n+1}}{(n+1)!} \implies |R_3(0.1)| \leq (1) \times \dfrac{|0.1-0|^{4}}{4!} \approx 4.2 \times 10^{-6}$ Now, $ \space Error \leq 4.2 \times 10^{-6}$
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