Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.1 Sequences - Exercises - Page 538: 40



Work Step by Step

We have $$ \lim\limits_{n \to \infty}{b_n}=\lim\limits_{n \to \infty}n^{1/n} .$$ By using the proerties of $\ln$ and L'Hopital's rule, we have $$\ln b_n=\frac{\ln n}{n} \Longrightarrow \lim_{n\to \infty} \ln b_n=\lim_{n\to \infty} \frac{\ln n}{n} =0.$$ Hence, $ \lim\limits_{n \to \infty}{b_n}=e^0=1.$ Using Theorem 1, we get that the sequence $b_n$ converges to $1$.
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