Consider the series $\sum_{n=1}^{\infty} a_n$ where $a_n = \frac{(2n-1)!!}{(2n)!!} \frac{1}{2n+1}$. Using Raabe's test, determine the convergence of the series by computing the limit $L = \lim_{n \to \infty} n \left(1 - \frac{a_{n+1}}{a_n}\right)$.