\Rightarrow \mathop {\lim }\limits_{n \to \infty } \frac{1}{{\left( {n + 1} \right)}}\frac{{n!}}{{n^{n + 1} }} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{\left( {n + 1} \right)}}\mathop {\lim }\limits_{n \to \infty } \sqrt {\frac{{2\pi }}{n}} e^{ - n} = 0