Math Help - Hyperfactorial function

1. Hyperfactorial function

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

2. Could we prove it by induction ? I think you must have a great method to solve it ( without induction )

3. Originally Posted by simplependulum
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 $\sum_{k=1}^n k \ln k.$

4. 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.