Prove that $\displaystyle \forall n \in \mathbb{N}: \ \ \prod_{k=1}^n k^k < (n+1)^{\frac{n(n+1)}{2}} e^{-\frac{(n-1)^2}{4}}.$

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- Nov 5th 2009, 03:01 AMNonCommAlgHyperfactorial function
Prove that $\displaystyle \forall n \in \mathbb{N}: \ \ \prod_{k=1}^n k^k < (n+1)^{\frac{n(n+1)}{2}} e^{-\frac{(n-1)^2}{4}}.$

- Nov 10th 2009, 12:05 AMsimplependulum
Could we prove it by induction ? I think you must have a great method to solve it ( without induction )

- Nov 10th 2009, 01:57 AMNonCommAlg
- Nov 15th 2009, 08:55 AMgirdav
Using logarithm, we have to show that $\displaystyle \frac{\left(n-1\right)^2}4\leq \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right)$.

We have $\displaystyle \sum_{k=1}^n k\left(\ln \left(n+1\right)-\ln \left(k\right)\right) = \sum_{k=1}^n k\ln \left( 1+\frac{n+1}k-1\right)$ and $\displaystyle \frac{n+1}k-1\geq 0$ so using $\displaystyle \ln\left(1+x\right) \geq x-\frac{x^2}2$ we should have the result.