Factorial Rational Limit

Question

Anything besides the squeeze theorem. Here it is:

$$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$

Can someone start me off?

Answer 1

By the ratio test

$$\frac{(2n+1)!}{(2n-1)!}\frac{n^n}{(n+1)^{n+1}}\sim4n\left(1+\frac1n\right)^{-n}\sim_\infty\frac4en\to\infty$$

so the given limit is $+\infty$.

Answer 2

L = $\lim_{n \to +\infty} \frac{a_{n+1}}{a_n}$. L is infinite therefore the limit of $a_n$is infinite.