Answer
Converges
Work Step by Step
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, $r=\lim\limits_{k \to \infty}\dfrac{a_{k+1}}{a_k}=\lim\limits_{k \to \infty}\dfrac{[(k+1)!]^3}{(3k+3)!} \times \dfrac{(3k)!}{(k!)^3} \\= \lim\limits_{k \to \infty} \dfrac{(k+1)^3}{(3k+3)(3k+2)(3k+1)}\\=\dfrac{1}{27} \lt 1$
Therefore, the series converges by the ratio test.
