Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.2 - Series - 11.2 Exercises - Page 716: 15


(a) convergent (b) divergent

Work Step by Step

(a) To determine whether $a_n$ converges, take the limit of $a_n$ as $n$ approaches $\infty$: $\lim\limits_{n \to \infty}a_n = \lim\limits_{n \to \infty}\frac{2n}{3n+1}=\frac{2}{3}$ since the numerator and denominator are of the same degree. Since $\lim\limits_{n \to \infty}a_n$ exists, $\{a_n\}$ is convergent. (b) From part (a), $\lim\limits_{n \to \infty}a_n =\frac{2}{3}$. By the Test for Divergence, since $\lim\limits_{n \to \infty}a_n \ne0$, the series $\Sigma_{n=1}^{\infty}a_n$ must diverge.
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