Calculus 8th Edition

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

Chapter 11 - Infinite Sequences and Series - Review - Exercises - Page 825: 39


$\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.

Work Step by Step

It is given that $\Sigma_{n=1}^{\infty} a_{n}$ is absolutely convergent , which means that the series $\Sigma_{n=1}^{\infty} |a_{n}|$ converges. Consider, $\Sigma_{n=1}^{\infty} |(\frac{n+1}{n})a_{n}|$ We have $\lim\limits_{n \to \infty}\frac{ |(\frac{n+1}{n})a_{n}|}{ |a_{n}|}=\lim\limits_{n \to \infty}|\frac{n+1}{n}|=1$ It follows that the given series behave like $\Sigma_{n=1}^{\infty} |a_{n}|$that is converges. So, we conclude that the series is $\Sigma_{n=1}^{\infty} (\frac{n+1}{n})a_{n}$ is absolutely convergent.
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