Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Consider a sequence $x_n, n\ge1$ formed by positive solutions to $x \sin{x}=1$.
How can we find
$$\lim _{n\rightarrow \infty}(n(x_{2n+1}-2\pi n))= L$$
and
$$\lim _{n\rightarrow \infty}(n^3(x_{2n+1}-2\pi n- \frac{L}{n}))= L_2$$
?