University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.1 - Sequences - Exercises - Page 488: 65

Answer

Divergent

Work Step by Step

Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{n!}{10^{6n}}$ Since, $\lim\limits_{n \to \infty} \dfrac{x^n}{n!}=0$ when $x \gt 0$ So, $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{n!}{10^{6n}}= \lim\limits_{n \to \infty} \dfrac{1}{\dfrac{(10^{6})^n}{n!}}=\dfrac{1}{0}=\infty$ Hence, $\lim\limits_{n \to \infty} a_n=\infty$ and {$a_n$} is Divergent
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