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Math Help - find the limit

  1. #1
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    find the limit

    \lim_{n->oo}\frac{(n!)^{1/n}}{n}
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  2. #2
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    it's a Riemann sum.

    first write a=e^{\ln a}.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    it's a Riemann sum.

    first write a=e^{\ln a}.
    sorry,i can not see it
    can u give me more idea??pls thx
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by chialin4 View Post
    \lim_{n->oo}\frac{(n!)^{1/n}}{n}
    It is commonly known that for convergent sequences the ration and root test gives the same result, in particular

    \lim_{n\to\infty}\frac{(n!)^{\frac{1}{n}}}{n}=\lim  _{n\to\infty}\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{  n^n}{n!}=\lim_{n\to\infty}\frac{n^n}{(n+1)^n}=\lim  _{n\to\infty}\left(1+\frac{1}{n}\right)^n=e
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