Are there any 'nice' expressions for
$$\arccos\frac2\pi$$
By 'nice', I mean involving rationals, square roots, powers of $\pi$, etc... I'm hoping there is something that doesn't involve sum or product series, nested radicals, continued fractions, integrals, derivatives, etc...
I'm aware of the Taylor series expansion for $\arccos$:
$$\arccos x=\frac\pi2-\sum_{k=0}^\infty{\frac{(2k)!x^{2k+1}}{2^{2k}(k!)^2(2k+1)}}$$
which for $x=\frac2\pi$ becomes
$$\arccos\frac2\pi=\frac\pi2-2\sum_{k=0}^\infty{\frac{(2k)!}{\pi^{2k+1}(k!)^2(2k+1)}}$$
Does the sum in this case converge to a 'nice' expression?