Problem 338 (difficulty: 5/10)

Prove that for all \(\displaystyle n\in \N\) we have \(\displaystyle \displaystyle n!>\left(\frac{n+1}e\right)^n\), and for \(\displaystyle n\ge7\) we have \(\displaystyle \displaystyle n!<\frac{n^{n+1}}{e^n}\).

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