I was looking for series that sum to $\pi$, and I happened to come across this one:
$\pi = \displaystyle\sum^{\infty}_{n=0}\frac{n!\left(2n\right)!\left(25n-3\right)}{2^{n-1}\left(3n\right)!}$
Could anyone please tell me why this series does indeed sum to $\pi$? It just seems odd.
We have to find the value of $$\begin{eqnarray*}\sum_{n\geq 1}\frac{n(50n-6)\,\Gamma(n)\,\Gamma(2n+1)}{2^n\,\Gamma(3n+1)}&=&\sum_{n\geq 1}\frac{n(50n-6)}{2^n}\int_{0}^{1}x^{n-1}(1-x)^{2n}\,dx\\&=&\int_{0}^{1}\frac{16\,(1-x)^2 \left(11+7 x-14 x^2+7 x^3\right)}{(2-x)^3 \left(1+x^2\right)^3}\,dx\end{eqnarray*}$$ that by partial fraction decomposition equals $\color{red}{\pi+6}$.
If you consider the original series over $n\geq 0$, you get $\pi$.