Long cycles in a random permutation

Let $C_n$ be the number of cycles longer than $n$ (i.e. of length $> n$) in a uniformly random permutation of $\{1, 2, \dots, 2n\}$. Compute $\displaystyle\lim_{n\to\infty}\mathbb{E}[C_n]$. The limit has the form $\ln(q)$ for a rational $q$; find $q$.