Answer
Converges
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$
1. For $l \lt 1$, the series is absolutely convergent.
2. For $l \gt 1$, the series is divergent.
3. For $l = 1$, the series is inconclusive.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty} \dfrac{k^{k+1}}{k^{k+1}}\\=\lim\limits_{k \to \infty} (\dfrac{k}{k+1})^k\\=\lim\limits_{k \to \infty} [1+\dfrac{1}{k})^k]^{-1} \\=\dfrac{1}{e} \lt 1$
So, we can conclude that the given series converges by the ratio test.