I tried to use D'Alambert theorem to determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$. $$\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} \frac{(2n-1)!!(2n+1)(2n)!!}{(2n-1)!!(2n)!!(2n+2)} = \lim_{n \to \infty} \frac{2n+1}{2n+2} = 1$$ but this test is inconclusive.
I think a comparison test might give a result, but with which series should I compare it to?