Show that the series $\sum\limits_{n\ge1}(n+1)^nn^{-n-3/2}$ converges Consider $\sum_{1}^{\infty} a_n$ , where $a_n$ =$\dfrac{(n+1)^n}{n^{n+\frac{3}{2}}}$
we find $$\dfrac{a_n}{a_{n+1}}=[\dfrac{(n+1)^2}{n(n+2)}]^n\times \dfrac{(n+1)^{\frac{5}{2}}}{(n+2)n^{\frac{3}{2}}}$$
by D-Alembert ratio test , $\lim_{n\rightarrow \infty}$ $\frac{a_n}{a_{n+1}}$ = $l$
i find $l= 1$, test fail,I have use Rabbes test, problem is very complicated, please help

Note that, for every positive $n$, $$(n+1)^n=\left(1+\frac1n\right)^n\cdot n^n\leqslant\mathrm e\cdot n^n$$ hence $$a_n\leqslant\frac{\mathrm e}{n^{3/2}}$$
