Calculus 8th Edition

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

Chapter 11 - Infinite Sequences and Series - 11.7 Strategy for Testing Series - 11.7 Exercises - Page 786: 33



Work Step by Step

Apply the Root Test: $\lim\limits_{n\to\infty}\sqrt[n]{|a_n|}$ $\lim\limits_{n\to\infty}\sqrt[n]{|a_n|}=\lim\limits_{n\to\infty}\sqrt[n]{\left(\frac{n}{n+1}\right)^{n^2}}$ $=\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^{\frac{n^2}{n}}$ $ =\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^{n}$ Divide the top and bottom by the highest power of $n$. $\lim\limits_{n\to\infty}\left(\frac{1}{1+\frac{1}{n}}\right)^{n}=\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}$ Since, $e=\lim\limits_{n\to \infty}(1+\frac{1}{n})^n$ Therefore, $\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}=\frac{1}{e}<1$ Hence, the series is convergent by the Root Test.
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