Show $n^{\frac{1}{n}}$ is decreasing for $n \ge 3$ How to show $\displaystyle n^{\frac{1}{n}}$ is decreasing for $n \ge 3$ ?

Late answer but $n>e$ so $$n^{1/n} > e^{1/n}>1 + 1/n = \frac{n+1}{n} \implies n^{1+1/n}>n+1 \implies n^{1/n} > (n+1)^{1/n+1}$$
