# Prove that the sequence is increasing

• January 13th 2012, 02:09 PM
alexmahone
Prove that the sequence is increasing
Prove that the sequence $a_n=\frac{n^n}{n!}$ is increasing.
• January 13th 2012, 02:22 PM
girdav
Re: Prove that the sequence is increasing
Compute $\frac{a_{n+1}}{a_n}$.
• January 13th 2012, 02:23 PM
Plato
Re: Prove that the sequence is increasing
Quote:

Originally Posted by alexmahone
Prove that the sequence $a_n=\frac{n^n}{n!}$ is increasing.

Can you show $\frac{{a_{n + 1} }}{{a_n }} = \left( {1 + \frac{1}{n}} \right)^n ~?$
• January 13th 2012, 02:37 PM
alexmahone
Re: Prove that the sequence is increasing
$a_{n+1}=\frac{(n+1)^{n+1}}{(n+1)!}=\frac{(n+1)^n}{ n!}>\frac{n^n}{n!}=a_n$

So, $a_n$ is increasing.