On the [Wolfram page about pi formulas](http://mathworld.wolfram.com/PiFormulas.html), there is this curious limit by R. W. Gosper (130) $$\lim\limits\_{n\to\infty}\prod\limits\_{k=n}^{2n}\dfrac{\pi}{2\arctan k}=4^{1/\pi}.$$


The only reference given is an entry from 1996 in some forum. Has anybody a proof or reference for this or similar formulas?



a limit by Gosper involving a product of arctan and $4^{1/\pi}$
Here is a sketch, which can be completed to a proof.


Asymptotically (i.e. for $k$ large), $\ln\left(\frac{\pi}{2\arctan k}\right) = \frac{2}{\pi k} + O(\frac{1}{k^2})$. Sum termwise to get 
$$\frac{2\Psi(2n+1)}{\pi} - \frac{2\Psi(n)}{\pi} + \sum\_{k=n}^{2n} O(\frac{1}{k^2})$$
Use the propreties of $\Psi$ to simplify this to
$$\frac{2\ln 2}{\pi} + \frac{\Psi(n+1/2)}{\pi} -\frac{\Psi(n)}{\pi} + \sum\_{k=n}^{2n} O(\frac{1}{k^2}) $$
Now, for $n$ large, this is asymptotically
$$\frac{2\ln 2}{\pi} + O(\frac{1}{n}).$$


All the terms in the product at positive, so taking the logarithm was legitimate. The termwise sum can similarly be justified.


