How to prove $\sqrt[n]{n}>\sqrt[n+1]{n+1}$ ,$n\ge 3$ monotone decreases by using this hint? I can solve it in other ways but I don't know how to solve it using this hint.
hints: Consider the expression $(1+n)^{n}\cdot n^{\frac{n+1}{n}}$ and use am-gm inequality.
note that$3^{\frac{-1}{3}}+\frac{1}{4}<1$