Convergence of a series defined with factorials This is the series in question.
$$\sum_{n=1}^{\infty} a_n \quad \text{where} \quad a_n = \frac{(2n)!}{4^nn!(n+1)!}$$
Naturally, I tried the Ratio test, but it turned out $L = 1$ so the test was inconclusive. In such series, if the Ratio test doesn't work, what can I try?

Hint:
By the duplication formula for Gamma function you have
$$
\left( {2\,n} \right)! = {{4^{\,n} } \over {\sqrt \pi  }}\left( {n - 1/2} \right)!n!
$$
