Convergence or divergence of a Sum

Question

I have this summation here which says:

I am asked to find if this sum converged or diverges. The only problem is that I am struggling with the denominator. What is denominator saying? It's just all the odd term being multiplied but I am not really sure how I would represent the denominator explicitly as a function or something.

How would I go about find if this series converges or diverges?

Answer 1

Hint: Observe \begin{align} 3\cdot 5 \cdots (2n+3) \geq 3\cdot 2^n\cdot n! \end{align} which means \begin{align} \sum^\infty_{n=1} \frac{n!}{3\cdot 5 \cdots (2n+3)} \leq \sum^\infty_{n=1}\frac{n!}{3\cdot 2^n\cdot n!}. \end{align}