just another $\pi$ formula
Precision such as the one employed in this expression is quite a bit overkill. One could simply
have asked for a proof of $~\displaystyle\lim_{n\to\infty}~\sum_{k=1}^n\dfrac{n^3~(2k-1)^2}{(n^2+k^2)^3}=\dfrac\pi8$ , which, come to think about it, looks
suspiciously similar to a simple Riemann sum...