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Thread: Limit of a Sequence

  1. #1
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    Limit of a Sequence

    Let the sequence an be :

    an = n!/n^n , and we know that : Lim n-------> infinity n!/n^n = 0 ,

    also we know that n! =
    Γ(n+1) ,

    Now an =
    Γ(x+1)/n^n ,

    Now , is it possible to find the limit of this sequence when n⇒ infinity , By using L'Hopital Rule by cosidering :


    f(x) =
    Γ(x+1)/x^x ,

    Lim x-------> infinity f(x) = ??? ( = 0 , by matlab)

    Note : I did this limit by matlab and the limit was : 0 ,

    I know that the derivative of the gamma function is : Γ(x)*psi(x) ,

    How could we do this in L'Hopital Rule and simplify that term to get zero finally ? ,


    Best regards
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  2. #2
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    Re: Limit of a Sequence

    $n!\le n^{n-1}$ for all $n\ge 1 $

    $0\le \dfrac{n!}{n^n} \le \dfrac{1}{n} $

    Use the Squeeze Theorem.
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  3. #3
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    Re: Limit of a Sequence

    Thanks for replay , you are right , but this way is already known , i am trying to find the limit by using L'Hopital Rule by converting n! to gamma(n+1) .
    Thanks again and best regards
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  4. #4
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    Re: Limit of a Sequence

    $\psi^{(0)}(x) = \dfrac{d}{dx}\left( \log \Gamma(x) \right) = \dfrac{\Gamma'(x)}{\Gamma(x)}$

    So, basically, you are finding that $\Gamma'(x) = \Gamma(x) \psi^{(0)}(x) = \Gamma'(x)$. That is not terribly useful. This is not in any way a simplification of the problem. Using polygamma functions gives more of the same.

    And taking successive derivatives of the denominator: $\dfrac{d}{dn}(n^n) = n^n(\ln n + 1)$ is going to give you an infinite series at the bottom that is similarly increasing in complexity rather than decreasing.
    Last edited by SlipEternal; Aug 27th 2017 at 10:55 AM.
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  5. #5
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    Re: Limit of a Sequence

    Now, something else you could do that might be a valid way of solving the problem:

    $\displaystyle \lim_{n \to \infty} \dfrac{n!}{n^n} = \lim_{n \to \infty} \prod_{k = 1}^n \dfrac{k}{n}$

    Now, you are looking to show that infinite product approaches zero. I am not sure if this is any easier.
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  6. #6
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    Re: Limit of a Sequence

    Dear SlipEternal

    Thank you soooo much for your replay and for your information , this is what i was looking for .

    Best regards
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