Answer
increasing
Work Step by Step
Given $$\left\{ \frac{n^n}{n!}\right\}_{1}^{\infty}$$
Since $a_n=\dfrac{n^n}{n!},\ \ a_{n+1}= \dfrac{(n+1)^{n+1}}{(n+1)!}$, then
\begin{align*}
\frac{a_{n+1}}{a_n}&=\dfrac{(n+1)^{n+1}}{(n+1)!}\dfrac{n!}{n^ n}\\
&= \dfrac{ (n+1)^{n}}{ n^n}\\
&=\left(1+\frac{1 }{ n }\right)^n \\
&\geq1
\end{align*}
So, the sequence is eventually increasing