TITLE: Convergence of infinite sum (factorial and root)
QUESTION [7 upvotes]: Find the convergence of the following sequence:
$$\sum_{n=1}^{\infty}\left(\dfrac{n!}{(2n)!}\right)^{\frac{1}{n}}$$
I tried to use the ratio test. It wasn't useful. Also I used the logharitmic test (I'm not sure this is the standard name for it): $\lim_{n\to\infty}\dfrac{\ln\frac{1}{u_n}}{\ln n}=l$. If $l>1$,     $\sum_{n=1}^{\infty}u_n$ converges. It didn't work to well. Thanks for your help!

REPLY [30 votes]: $$\left(\frac{n!}{(2n)!}\right)^{1/n}=\left(\frac1{(n+1)(n+2)\cdots(2n)}\right)^{1/n}>\left(\frac1{(2n)(2n)\cdots(2n)}\right)^{1/n}=\frac1{2n}$$
hence the series diverges.