How to prove that this infinite series converges

Question

The series in question is in the expression below. I know that the series converges but I am not sure how to prove so. I feel like there might be an infinite product involved, but we haven’t been taught that in my calculus class yet, so is there a different way to do it?

$$\sum_{n=0}^{\infty}\frac{n!}{2\cdot 5\cdot 8\cdots(3n+2)}.$$

Answer 1

Hint. Consider the limit of the ratio $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{n+1}{3(n+1)+2}.$$ and use the ratio test.

Answer 2

Since $$\frac{k}{3k+2}<\frac{1}{2},$$ we obtain: $$\sum_{k=0}^{+\infty}\frac{k!}{2\cdot...\cdot(3k+2)}<\sum_{k=0}^{+\infty}\frac{1}{2^{k+1}}=1$$