The goal of this exercise is to establish Formula (5), namely,limn+xnn! = 0Let an =

Chapter 9, Problem 31

(choose chapter or problem)

The goal of this exercise is to establish Formula (5), namely,

\(\lim \limits_{n \rightarrow+\infty} \frac{x^{n}}{n !}=0\)

Let \(a_{n}=|x|^{n} / n !\) and observe that the case where \(x=0\) is obvious, so we will focus on the case where \(x \neq 0\)
(a) Show that

\(a_{n+1}=\frac{|x|}{n+1} a_{n}\)

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

(c) Show that the sequence \(\left\{a_{n}\right\}\) converges.

Equation Transcription:

Text Transcription:

lim_n right arrow +infinity x^n /n! = 0

a_n = |x|^n /n!

x=0

x neq 0

a_n+1 = |x| /n+1 a_n

{a_n}