Answer
Inconclusive
Work Step by Step
Let us suppose a series $a_k$ such that $L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$ converges when $L\lt 1$ and diverges when $L \gt 1$.
Now, we have: $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty} \dfrac{(k+1)^{k+1}}{(k+1)!} \times \dfrac{k^k}{k!}\\=\lim\limits_{k \to \infty} |\dfrac{-(k+1)^k}{k^k}|$
Use L-hospitals Rule: $L=\lim\limits_{k \to \infty} |\dfrac{k !}{k !}|\\=1$
So, we can conclude that the limit is inconclusive.
