$$\sum\limits_{k = 0}^{ + \infty } {{{\left( { - 1} \right)}^k}{{\left( {\prod\limits_{i = 1}^k {\frac{{2i - 1}}{{2i}}} } \right)}^2}} = \frac{1}{{2\pi }}{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right) = \frac{{\sqrt {2\pi } }}{{2\Gamma {{\left( {\frac{3}{4}} \right)}^2}}}.$$