# Math Help - Prove that the sequence is increasing

1. ## Prove that the sequence is increasing

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

2. ## Re: Prove that the sequence is increasing

Compute $\frac{a_{n+1}}{a_n}$.

3. ## Re: Prove that the sequence is increasing

Originally Posted by alexmahone
Prove that the sequence $a_n=\frac{n^n}{n!}$ is increasing.
Can you show $\frac{{a_{n + 1} }}{{a_n }} = \left( {1 + \frac{1}{n}} \right)^n ~?$

4. ## 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.