I have a problem with the following sequence $ \lim_{n \to \infty} g_n \stackrel{?}{=} \pi $ where $g_n = \sum_{k=1}^{n-1} \frac{\sqrt{\frac{2n}{k}-1}}{n-k} + \sum_{k=n+1}^{2n-1}\frac{\sqrt{\frac{2n}{k}-1}}{n-k}.$
Does it converge to $\pi$? I tested experimentally that it does, but I was unable to prove it by hand. Could anybody help, or offer some methods of approach?