Use the Ratio Test to test for convergence or divergence of the series

Question

$$\sum_{n=1}^\infty\frac{(2n)!}{n^5}$$

I applied the Ratio Test and ended up with $[(2n+2)!(n^5)]/[(n+1)^5(2n)!]$ which I simplified down: $\lim_{n\to\infty}2n^5(2n+1)/(n+1)^4 = \infty > 1$ so it diverges

Is this correct? (Sorry about the formatting)

Answer 1

Yes but a very big gun for this series. Note that $$ \lim_{n\rightarrow \infty}\frac{(2n)!}{n^5}=\infty $$ and all convergent series must have the limit of their terms tend to $0$ (although beware, this is no guarantee).