tag:
Рамануджан
74
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Формула Рамануджана для числа \[\pi \]

\[\frac{{2\sqrt 2 }}{{9801}}\sum\limits_{n = 0}^{ + \infty } {\frac{{\left( {4n} \right)!\left( {1103 + 26390n} \right)}}{{{{\left( {n!} \right)}^4}{{396}^{4n}}}}} = \frac{1}{\pi }.\]
77
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[Рамануджан]



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



Примечание

\[\begin{array}{l} S = \sum\limits_{k = 1}^{ + \infty } {\frac{1}{{\left( {2k - 1} \right)!!}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \left( {1 - \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right)} \right) \hfill \\ F = \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + ...}}}}}}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right) \hfill \\ \operatorname{erfc} x = 1 - \operatorname{erf} x \hfill \\ \operatorname{erf} x = \frac{2}{{\sqrt \pi }}\int\limits_0^x {{e^{ - {t^2}}}dt} \hfill \\ \end{array}\]
1410.
\[\begin{array}{l} \sqrt[3]{{\cos \frac{{2\pi }}{7}}} + \sqrt[3]{{\cos \frac{{4\pi }}{7}}} + \sqrt[3]{{\cos \frac{{8\pi }}{7}}} = \sqrt[3]{{\frac{{5 - 3\sqrt[3]{7}}}{2}}} \hfill \\ \sqrt[3]{{\cos \frac{{2\pi }}{9}}} + \sqrt[3]{{\cos \frac{{4\pi }}{9}}} + \sqrt[3]{{\cos \frac{{8\pi }}{9}}} = \sqrt[3]{{\frac{{3\sqrt[3]{9} - 6}}{2}}} \hfill \\ \end{array}\]
1888.
\[\begin{array}{l} \left[ {{\text{Рамануджан}}} \right] \hfill \\ \sqrt[3]{{\sqrt[3]{2} - 1}} = \sqrt[3]{{\frac{1}{9}}} - \sqrt[3]{{\frac{2}{9}}} + \sqrt[3]{{\frac{4}{9}}} \hfill \\ \end{array}\]
1890.
\[\begin{array}{l} \left[ {{\text{Рамануджан}}} \right] \hfill \\ \sqrt[4]{{\frac{{3 + 2\sqrt[4]{5}}}{{3 - 2\sqrt[4]{5}}}}} = \frac{{\sqrt[4]{5} + 1}}{{\sqrt[4]{5} - 1}} \hfill \\ \end{array}\]