Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.9 Representations of Functions as Power Series - 11.9 Exercises - Page 797: 17


$\sum_{n=0}^{\infty}(-1)^{n}{4^{n}}(n+1)x^{n+1}$, $R=\frac{1}{4}$

Work Step by Step

$f(x)=\frac{x}{(1+4x)^{2}}=\sum_{n=0}^{\infty}(-1)^{n}{4^{n}}(n+1)x^{n+1}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{(-1)^{n+1}{4^{n+1}}(n+2)x^{n+2}}{(-1)^{n}{4^{n}}(n+1)x^{n+1}}|$ $=|x|\lt \frac{1}{4}$ The given series converges with $R=\frac{1}{4}$
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