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: 54



Work Step by Step

Given $$ \sum_{n=1}^{\infty} \frac{n^2+4n}{3n^4+9}$$ Compare with the convergent series $ \sum_{n=1}^{\infty} \frac{1}{n^2}$ ($p-$series, $p>1$); then by using the limit comparison test, we get: \begin{align*} \lim_{n\to \infty} \frac{a_n}{b_n}&=\lim_{n\to \infty}\frac{n^4+4n^3}{3n^4+9}\\ &=\lim_{n\to \infty}\frac{1+4/n}{3 +9/n^4}\\ &=\frac{1}{3} \end{align*} Thus the given series also converges.
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