Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.3 - The Integral Test - Exercises 10.3 - Page 586: 25

Answer

Divergent

Work Step by Step

Let us consider $a_n=\lim\limits_{n \to \infty} \dfrac{2^n}{n+1}$ Here, $\lim\limits_{n \to \infty} \dfrac{2^n}{n+1}=\dfrac{\infty}{\infty}$ This shows that the limit has an indeterminate form so, we will use L-Hospital's rule. so, $\lim\limits_{n \to \infty} \dfrac{(2^n) (\ln 2)}{1} =\infty $ Thus, the series is divergent.
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