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Math Help - Inequality with e, factorial, exponential function...

  1. #1
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    Inequality with e, factorial, exponential function...

    Hi!


    Could you help me prove this inequality?
    e^{n-1} \cdot  n! < n^{n+1}

    When I try induction , I get:
    e^n \cdot  (n+1)! = e^{n-1} \cdot  n!  \cdot e \cdot (n+1)<

    and this is when I have no idea how to get (n+1)^{n+2}. Could I ask for a small hint? It cannot be that hard to prove
    Or maybe you have another, more constructive way to prove this inequality?
    Last edited by Plato; December 9th 2012 at 05:06 AM.
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    Forum Admin topsquark's Avatar
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    Re: Inequality with e, factorial, exponential function...

    Quote Originally Posted by wilhelm View Post
    Hi!


    Could you help me prove this inequality?
    e^{n-1} \cdot  n! < n^{n+1}

    When I try induction , I get:
    e^n \cdot  (n+1)! = e^{n-1} \cdot  n!  \cdot e \cdot (n+1)<

    and this is when I have no idea how to get [latex](n+1)^{n+2}[/latex]. Could I ask for a small hint? It cannot be that hard to prove
    Or maybe you have another, more constructive way to prove this inequality?
    Made an oopsie!

    -Dan
    Last edited by topsquark; December 9th 2012 at 03:22 AM.
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Inequality with e, factorial, exponential function...

    Dan, I have found the inequality is correct for 1<n\in\mathbb{N} and provided a proof by induction on another site.

    For example, for n=2, we have:

    2e<8

    and for n=3 we have:

    6e^2<3^4

    -Mark
    Thanks from topsquark
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  4. #4
    Forum Admin topsquark's Avatar
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    Re: Inequality with e, factorial, exponential function...

    Quote Originally Posted by MarkFL2 View Post
    Dan, I have found the inequality is correct for 1<n\in\mathbb{N} and provided a proof by induction on another site.

    For example, for n=2, we have:

    2e<8

    and for n=3 we have:

    6e^2<3^4

    -Mark
    I worked out a few on my calculator for convenience and used "e" instead of Exp.

    Thanks for the catch.

    -Dan
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