Answer
Diverges
Work Step by Step
We are given that $a_k=\dfrac{k^k}{2^k}$
Ratio Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\dfrac{a_{k+1}}{a_k}$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{k \to \infty}\dfrac{(k+1)^{k+1} 2^k}{k^k 2^{(k+1)}}\\=\lim\limits_{k \to \infty} \dfrac{(k+1)^k (k+1)}{2k^k}\\=\lim\limits_{k \to \infty} \dfrac{(1+\dfrac{1}{k})^k(k+1)}{2}\\=\infty$
Therefore, the series diverges by the ratio test.
