Prove that the sequence is increasing

**alexmahone** · January 13th 2012, 01:09 PM

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

**girdav** · January 13th 2012, 01:22 PM

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

**Plato** · January 13th 2012, 01:23 PM

Originally Posted by alexmahone

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

Can you show $\frac{{a_{n + 1} }}{{a_n }} = \left( {1 + \frac{1}{n}} \right)^n ~?$

**alexmahone** · January 13th 2012, 01:37 PM

$a_{n+1}=\frac{(n+1)^{n+1}}{(n+1)!}=\frac{(n+1)^n}{ n!}>\frac{n^n}{n!}=a_n$

So, $a_n$ is increasing.

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