Chapter 9.2, Problem 31ES

The goal of this exercise is to establish Formula (5), namely, $\underset{n\to +\infty }{\lim }\frac{x^n}{n!}=0$

Let $a_n={\left|x\right|}^n/n!$ and observe that the case where $x=0$ is obvious, so we will focus on the case where $x\ne 0.$

(a) Show that $a_n+1=\frac{\left|x\right|}{n+1}{a}_n$

(b) Show that the sequence $\left\{a_n\right\}$ is eventually strictly decreasing.

(c) Show that the sequence $\left\{a_n\right\}$ converges.