\[\frac{2}{\pi } = \frac{{\sqrt 2 }}{2} \cdot \frac{{\sqrt {2 + \sqrt 2 } }}{2} \cdot \frac{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }}{2}...\]

\[\frac{{2\sqrt 2 }}{{9801}}\sum\limits_{n = 0}^{ + \infty } {\frac{{\left( {4n} \right)!\left( {1103 + 26390n} \right)}}{{{{\left( {n!} \right)}^4}{{396}^{4n}}}}} = \frac{1}{\pi }.\]

\[\frac{\pi }{2} = \prod\limits_{n = 1}^{ + \infty } {\frac{{4{n^2}}}{{4{n^2} - 1}}} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdot ...\]

\[1 + \frac{1}{{1 \cdot 3}} + \frac{1}{{1 \cdot 3 \cdot 5}} + \frac{1}{{1 \cdot 3 \cdot 5 \cdot 7}} + \frac{1}{{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}} + \cdot \cdot \cdot + \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + \frac{4}{{1 + ...}}}}}}}}}} = \sqrt {\frac{{\pi \cdot e}}{2}} .\]



Примечание

\[\begin{array}{l} S = \sum\limits_{k = 1}^{ + \infty } {\frac{1}{{\left( {2k - 1} \right)!!}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \left( {1 - \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right)} \right) \hfill \\ F = \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + ...}}}}}}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right) \hfill \\ \operatorname{erfc} x = 1 - \operatorname{erf} x \hfill \\ \operatorname{erf} x = \frac{2}{{\sqrt \pi }}\int\limits_0^x {{e^{ - {t^2}}}dt} \hfill \\ \end{array}\]