$$\mathop {\lim }\limits_{n \to \infty } \left( {n \cdot \prod\limits_{k = 1}^n {\frac{{1 + {{\left( {2k} \right)}^2}}}{{1 + {{\left( {2k + 1} \right)}^2}}}} } \right) = \tanh \frac{\pi }{2}$$