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: 16


$a_n$ converges to $0$.

Work Step by Step

We have $$ \lim _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} \frac{100^{n}}{n !}-\frac{3+\pi^{n}}{5^{n}} \\ =\lim _{n \rightarrow \infty} \frac{100^{n}}{n !}-\lim _{n \rightarrow \infty} \frac{3}{ 5^{n}}- \lim _{n \rightarrow \infty} (\frac{\pi}{5})^{n} =0. $$ Where we used the facts: for large $n\gt \gt 100$ , $n!$ is getting bigger faster than $100^n$; also, $\pi/5\lt 1$. Thus, $a_n$ converges to $0$.
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