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Math Help - Hyperfactorial function

  1. #1
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    Hyperfactorial function

    Prove that \forall n \in \mathbb{N}: \ \ \prod_{k=1}^n k^k < (n+1)^{\frac{n(n+1)}{2}} e^{-\frac{(n-1)^2}{4}}.
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  2. #2
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    Could we prove it by induction ? I think you must have a great method to solve it ( without induction )
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  3. #3
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    Quote Originally Posted by simplependulum View Post
    Could we prove it by induction ? I think you must have a great method to solve it ( without induction )
    induction would be fine if you could finish it. another way is to use Euler's summation formula to find an upper bound for \sum_{k=1}^n k \ln k.
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    Using logarithm, we have to show that \frac{\left(n-1\right)^2}4\leq \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right).
    We have \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right) = \sum_{k=1}^n k\ln \left( 1+\frac{n+1}k-1\right) and \frac{n+1}k-1\geq 0 so using  \ln\left(1+x\right) \geq x-\frac{x^2}2 we should have the result.
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