Answer
Converges
Work Step by Step
We are given that $a_k=\dfrac{k^5}{e^k}$
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, $r=\lim\limits_{k \to \infty} |\dfrac{2^{k+1} k!}{2^k (k+1)!}|\\=\lim\limits_{k \to \infty} \dfrac{\dfrac{(k+1)^5}{e \cdot e^k}}{\dfrac{k^5}{e^k}}\\=\dfrac{1}{e} \lim\limits_{k \to \infty} (\dfrac{k+1}{k})^5\\=\dfrac{1}{e} (1)^5 \\=\dfrac{1}{e} \lt 1$
Therefore, the series converges by the ratio test.
