Proof of an inequality involving factorials

Question

How can the following inequality be proven? $$\left(n!\right)^{\frac{1}{n}}\left((n+1)!\right)^{-\frac{1}{n+1}}\gt\dfrac{n}{n+1}$$ I know this is a result obtained in 1964, but I don't know how to prove it or where to find the proof. Thanks

Answer 1

Here is the sketch of the proof. $$ (n!)^{n+1}((n+1)!)^{-n}\gt \frac{n^{n^2+n}}{(n+1)^{n^2+n}},\\ \frac{n!}{(n+1)^n}\gt \frac{n^{n(n+1)}}{(n+1)^{n(n+1)}},\\ n!\gt \frac{n^{n(n+1)}}{(n+1)^{n^2}},\\ $$ Last inequality can be proved using induction and $$ \left(1+\frac{1}{n}\right)^n \lt \left(1+\frac{1}{n+1}\right)^{n+1}. $$