# limits problem

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• Jun 30th 2008, 06:32 PM
omrimalek
limits problem
Halllo

I need to proove (in a formal way) in a formal way that when n=>infinity, lim (n^n)/((n!)^2)=0

if someone can hekp i would relly appriciate it...

Omri
• Jun 30th 2008, 07:00 PM
NonCommAlg
Quote:

Originally Posted by omrimalek
Halllo

I need to proove (in a formal way) in a formal way that when n=>infinity, lim (n^n)/((n!)^2)=0

if someone can hekp i would relly appriciate it...

Omri

let $a_n=\frac{n^n}{(n!)^2}.$ then: $\frac{a_{n+1}}{a_n}=\frac{1}{n+1} \left(1 + \frac{1}{n} \right)^n.$ thus: $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 0 < 1.$ hence $\sum a_n$ is convergent. so: $\lim_{n\to\infty}a_n=0. \ \ \ \square$
• Jun 30th 2008, 07:51 PM
Mathstud28
Quote:

Originally Posted by omrimalek
Halllo

I need to proove (in a formal way) in a formal way that when n=>infinity, lim (n^n)/((n!)^2)=0

if someone can hekp i would relly appriciate it...

Omri

NonCommonAlg's way is fine, but if you haven't done series yet I would use Stirlings approximation.

The two ways to use it would be to use substitution or use it as a mean of finding two bounding functions for sufficently large n and go the Squeeze Theorem route.