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?

Use this theorem

Let $\{a_n\} $ and $\{b_n\}$ be two sequences with $a_n>0$ and $b_n>0 $ for all $n \geq N$ and let $c_n = b_n - b_{n+1} \frac{a_{n+1}}{a_n}$ then if there is a positive constant $r$ such that $c_n \geq r > 0$ for all $n \geq N $ then $\sum a_n$ converges

with your series (simplifying you obtain $\frac{a_{n+1}}{a_n} = \frac{2n+1}{2(n+2)} $) and $b_{n+1} = n+2$ so $$c_n = n+1-(n+2)\frac{2n+1}{2(n+2)} = \frac{1}{2}$$
therefore, we can use $r = \frac12$ to show that $\sum a_n $ converges.
