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: 15


$ln5-\sum_{n=1}^{\infty}\frac{x^{n}}{5^{n}n}$ $R=5$

Work Step by Step

$f(x)=ln(5-x)=ln5-\sum_{n=0}^{\infty}\frac{x^{n+1}}{5^{n+1}n+1}=ln5-\sum_{n=1}^{\infty}\frac{x^{n}}{5^{n}n}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{x^{(n+1)+1}}{5^{n+1}n+1}}{\frac{x^{n+1}}{5^{(n+1)+1}(n+1)+1}}|$ $=|\frac{x}{5}|\lt 1$ The given series converges with $R=5$
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