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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.

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    Now I found this older question, which seems to be about the same thing: http://math.stackexchange.com/questions/788096/how-to-prove-that-lim-n-to-infty-frac2n-12n-0 – Martin Sleziak Dec 01 '15 at 15:11

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