Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 20


$c_n$ converges to $e^3$.

Work Step by Step

We have $$ \lim _{n \rightarrow \infty} c_n=\lim _{n\rightarrow \infty} \left(1+\frac{3}{n}\right)^{n} \\ =\lim _{n\rightarrow \infty} \left(1+\frac{1}{n/3}\right)^{n} =\left(\lim _{n/3\rightarrow \infty} \left(1+\frac{1}{n/3}\right)^{n/3}\right)^{3}=e^3. $$ Where we used the fact that $\lim _{m \rightarrow \infty} \left(1+\frac{1}{m}\right)^{m}=e$. So, $c_n$ converges to $e^3$.
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