tag:
бесконечные_произведения
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Формула Виета для числа \[\pi \]
\[\frac{2}{\pi } = \frac{{\sqrt 2 }}{2} \cdot \frac{{\sqrt {2 + \sqrt 2 } }}{2} \cdot \frac{{\sqrt {2 + \sqrt {2 + \sqrt 2 } } }}{2}...\]
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Формула Гаусса-Якоби
$$\prod\limits_{k = 1}^{ + \infty } {\left( {1 + {u^{k - 1}}{v^k}} \right)\left( {1 + {u^k}{v^{k - 1}}} \right)\left( {1 - {u^k}{v^k}} \right)} = \sum\limits_{q = - \infty }^{ + \infty } {{u^{\frac{{q\left( {q + 1} \right)}}{2}}}{v^{\frac{{q\left( {q - 1} \right)}}{2}}}} $$
2116.
$$\mathop {\lim }\limits_{n \to \infty } \left( {n \cdot \prod\limits_{k = 1}^n {\frac{{1 + {{\left( {2k} \right)}^2}}}{{1 + {{\left( {2k + 1} \right)}^2}}}} } \right) = \tanh \frac{\pi }{2}$$
2139.
$$\prod\limits_{k = 1}^{ + \infty } {\frac{{2k\left( {4k + 3} \right)}}{{\left( {2k + 1} \right)\left( {4k + 1} \right)}}} = \frac{1}{6}{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right)$$
2149.
$$\eqalign{
& \prod\limits_{k = 1}^{ + \infty } {{{\left( {1 + \frac{1}{{2k}}} \right)}^{{{\left( { - 1} \right)}^k}}}} = \frac{4}{{{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right)}} \cr
& \prod\limits_{k = 1}^{ + \infty } {{{\left( {1 - \frac{1}{{2k}}} \right)}^{{{\left( { - 1} \right)}^k}}}} = \frac{1}{\pi }{\rm B}\left( {\frac{1}{2},\frac{1}{4}} \right) \cr
& \prod\limits_{k = 1}^{ + \infty } {\left( {1 + \frac{{{{\left( { - 1} \right)}^{k + 1}}}}{{2k}}} \right)} = \frac{1}{{2\pi }}{\rm B}\left( {\frac{1}{4},\frac{1}{4}} \right) \cr} $$
2150.
$$\prod\limits_{k = 1}^{ + \infty } {\frac{{2k\left( {2k + 2} \right)}}{{{{\left( {2k + 1} \right)}^2}}}} = \frac{\pi }{4}$$
2151.
$$\prod\limits_{k = 1}^{ + \infty } {\frac{{{{\left( {3k + 2} \right)}^3}}}{{27k{{\left( {k + 1} \right)}^2}}}} = {\left( {\frac{3}{{2\Gamma \left( {\frac{2}{3}} \right)}}} \right)^3}$$