After applying the Ratio Test we can cancel most terms in the factorials. But what to do with the denominator in the original series? Can it be considered as a factorial?
Also, I think we can use the divergence test in this series, as I think the limit will approach $\infty$ as $n$ approaches $\infty$.
Yes, you have indeed$$\lim_{n\to\infty}\frac{(2n)!}{3\times5\times\cdots\times(2n+1)}=\infty,$$but you have to prove it.
On the other hand,$$\frac{\frac{(2(n+1))!}{3\times5\times\cdots\times(2n+3)}}{\frac{(2n)!}{3\times5\times\cdots\times(2n+1)}}=\frac{(2n+1)(2n+2)}{2n+3}$$and, since $\lim_{n\to\infty}\frac{(2n+1)(2n+2)}{2n+3}$ diverges indeed by the ratio test.