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 10.7 - Power Series - Exercises 10.7 - Page 613: 39



Work Step by Step

Let us appy the Ratio Test to the given series. In order to find the radius of convergence, we have: $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_n}|=|\lim\limits_{n \to \infty} \dfrac{\dfrac{(n+1)a_nx}{4(2n+1)}}{a_n}| \\=|x| \lim\limits_{n \to \infty} \dfrac{1+\dfrac{1}{n}}{8+(4/n)} \\=\dfrac{1}{8}|x|$ Now, $\dfrac{1}{8}|x| \lt 1$ and $|x| \lt 8$ Thus, the radius of convergence is $8$
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