Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - Exercises 11.7.1 - 11.7.9: 1

Answer

The series converges by the Comparison Test.

Work Step by Step

Use the Comparison Test. Let $a_n=\frac{1}{n+3^n}$ and $ b_n=\frac{1}{3^n}$. The series $\frac{1}{3^n}>\frac{1}{n+3^n}$ for all $n>1$. $\frac{1}{3^n}=\left(\frac{1}{3}\right)^n$ This is a geometric series with $[|r|=\frac{1}{3}<1]$; therefore, the series is convergent. Because $\sum_{n=1}^\infty \frac{1}{3^n}$ is convergent and greater than $\sum_{n=1}^{\infty}\frac{1}{n+3^n}$ for all $n>1$ the series $\sum_{n=1}^{\infty}\frac{1}{n+3^n}$ is also convergent.
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