I have tried using Stolz–Cesàro's formula and subtract the next term in the series but that gave me $$\lim_{n\to \infty} {\frac{2n}{2n+1}}$$ which is obviously 1 and not right. I do realise that the limit should be 0 but I don't know how to prove it.
Edit due to comment:
$n!!$ is the double factorial, as explained in this wolfram post.