\frac{(2n)!!}{(2n+1)!!}=\frac{4^n(n!)^2}{(2n+1)!}>\frac{1}{\sqrt{2n+1}}\frac{4^n2\pi n^{2n+1}e^{-2n+\frac{1}{6n}-\frac{1}{180n^3}}}{\sqrt{2\pi}(2n+1)^{2n+1}e^{-2n-1+\frac{1}{24n+12}}}=\frac{1}{\sqrt{2n+1}}\sqrt{\frac{\pi}{2}}\frac{e}{(1+\frac 1{2n})^{2n+1}}e^{\frac{1}{6n}-\frac{1}{24n+12}-\frac{1}{180n^3}}