Find the nature of $\sum_{n = 1}^\infty \frac{(2n - 1)!!}{(2n)!! (2n + 1)}$

Question

I need to find whether the following series converges or diverges:

$$\sum_{n = 1}^\infty \frac{(2n - 1)!!}{(2n)!! (2n + 1)}$$

It seems to converge but I don't really have any good idea on how to prove it. I tried to use D'Alembert Criterion, but the limit is $1$, so it doesn't help. Can you give me a hint, please?

Answer 1

$n(\frac{a_n}{a_{n+1}}-1)\to\frac{3}{2}>1$ as $n\to\infty$ and it converges by Raabe's test.

Answer 2

You can also just note that \begin{align*} \frac{(2n-1)!!}{(2n)!! (2n+1)}& = \frac{1\times 2 \times \cdots \times (2n-1)!}{(2n+1) \times 1 \times 2 \times \cdots \times (2n-1)! \times ((2n-1)!+1)\times \cdots \times (2n)!}\\ = & \frac{1}{(2n+1)\times [(2n-1)!+1] \times [(2n-1)!+2] \times \cdots (2n)!} \\ < & \frac{1}{2n^2} \end{align*}

and that $\sum \frac{1}{2n^2}$ is convergent.