The ratio test with a parameter

Ratio.
$$\frac{u_{n+1}}{u_n}=a\cdot\frac{(n+1)\,n^n}{(n+1)^{n+1}}=a\left(1+\frac1n\right)^{-n}\longrightarrow\frac ae.$$

Cases. For $a<e$ the series converges; for $a>e$ it diverges.

Critical case $a=e$. By Stirling,
$$u_n=\frac{n!\,e^n}{n^n}\sim\sqrt{2\pi n}\longrightarrow+\infty,$$
so the terms do not tend to $0$: the series diverges.

Conclusion. The series converges iff $0<a<e$.