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Math Help - limits problem

  1. #1
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    Smile 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
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  2. #2
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    Quote Originally Posted by omrimalek View Post
    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
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by omrimalek View Post
    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.
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