Answer
The series converges.
Work Step by Step
We are given that $a_k=\dfrac{2^k}{k!}$
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, $r=\lim\limits_{k \to \infty}\dfrac{2^{k+1} k!}{2^k (k+1)!}\\=\lim\limits_{k \to \infty} \dfrac{2 \cdot 2^ k \cdot k!}{2^k \cdot k!(k+1)}\\=\lim\limits_{k \to \infty} \dfrac{2}{k+1}\\=0$
Therefore, the series converges by the ratio test.
