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Math Help - Integral Inequality Proof

  1. #1
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    Integral Inequality Proof

    Prove:

    ln({n!}) < \int_{1}^{n+1} \ln(x) dx
    Last edited by Jameson; September 9th 2009 at 07:27 AM. Reason: fixed latex
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  2. #2
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    Quote Originally Posted by Paul616 View Post
    Prove:

    ln({n!}) < \int_{1}^{n+1} \ln(x) dx
    Prove
     \ln k = \int_{k}^{k+1} \ln k \ dx < \int_{k}^{k+1} \ln x \ dx

     \sum \ln k = \sum \int_{k}^{k+1} \ln x \ dx
    Last edited by mr fantastic; September 18th 2009 at 07:58 AM. Reason: Restored original post
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  3. #3
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    Quote Originally Posted by Paul616 View Post
    Prove:

    ln({n!}) < \int_{1}^{n+1} \ln(x) dx
    Notice that \ln n! = \sum_{k=1}^n \ln k.
    Consider the function f(x) = \ln x.
    This function is positive and decreasing for x>0.
    Therefore, the sum of rectangular areas [1,n] by using right-end points is smaller than the area.

    This is basically all you need to argue.
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