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 634: 56


Diverges for $|x| \gt 1$.

Work Step by Step

$\lim\limits_{n \to \infty}|\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(-1)x^2 (2n-1)}{2n+1}| $ or, $=x^2 \lim\limits_{n \to \infty} \dfrac{2-1/n1}{2+1/n}$ or, $=x^2$ Now, $| Error|=|\dfrac{(-1)^n x^n}{n}|=\dfrac{1}{n (10^n)}$ But $\dfrac{1}{2n-1} \lt 10^{-3}$ or, $\dfrac{1}{2n-1} \lt \dfrac{1}{10^{3}} $ or, $ n=501$ Thus, the series is divergent for $|x| \gt 1$.
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