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 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 54


7 terms

Work Step by Step

Recall the Taylor series for: $\ln (1+x)=x-\dfrac{ x^2}{2}+\dfrac{x^3}{3}-....$ ; $-1 \leq x \leq 1$ and $\ln (1-x)=-x-\dfrac{ x^2}{2}-\dfrac{x^3}{3}-....$ ; $-1 \leq x \leq 1$ Now, we have $| Error|=|\dfrac{(-1)^n x^n}{n}|$ Set $ x=0.1$; then we have $| Error|=|\dfrac{(-1)^n x^n}{n}|=\dfrac{1}{n (10^n)}$ But $\dfrac{1}{n (10^n)} \lt \dfrac{1}{10^8}$ This implies that $ n \geq 8$ Thus, we need $7$ terms for the accuracy.
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