Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 29

Answer

Converges

Work Step by Step

Given $$\sum_{n=1}^{\infty}n^3a_n,\ \ \ \ |a_{n+1}/a_n| =\frac{1}{3}$$ By using the Ratio test, let $b_n= n^3a_n$ \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{b_{n+1}}{b_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|\frac{(n+1)^3a_{n+1}}{n^3a_n} \right|\\ &= \lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^3 \frac{a_{n+1}}{a_n} \\ &= \lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^3 \lim _{n \rightarrow \infty}\frac{a_{n+1}}{a_n} \\ &= \frac{1}{3}<1 \end{align*} Thus the series converges.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.