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Eariu52

\(\displaystyle{ \frac{ \sqrt{3} \pi }{2} = \sum_{n=0}^{ \infty } \frac{\left( -1\right) ^{n} }{\left( n+ \frac{1}{3} \right)\left( n+1\right) } }\)

Eariu52

\(\displaystyle{ 6 \sqrt{3} \pi = \sum_{n=0}^{ \infty } \frac{1}{\left( n+ \frac{1}{6} \right)\left( n+ \frac{2}{6} \right) \left( n+ \frac{4}{6} \right) \left( n+ \frac{5}{6} \right) } }\)

Eariu52

\(\displaystyle{ \frac{9 \pi }{2} = \sum_{n=0}^{ \infty } \frac{\left( -1\right) ^{n} }{\left( n + \frac{1}{6} \right) \left( n + \frac{3}{6} \right) \left( n + \frac{5}{6} \right) } }\)

Eariu52

\(\displaystyle{ \frac{8 \pi }{3}= \sum_{n=0}^{ \infty } \frac{1}{\left( n+ \frac{1}{4} \right) \left( n+ \frac{1}{2} \right)\left( n+1\right) } }\)

Eariu52

\(\displaystyle{ \sqrt{3} = \frac{ \sqrt{2}+1 }{ \sqrt{2} } \cdot \frac{ \sqrt{2- \sqrt{2} }+1 }{ \sqrt{2+ \sqrt{2} } } \cdot \frac{ \sqrt{2- \sqrt{2- \sqrt{2} } }+1 }{ \sqrt{2+ \sqrt{2+ \sqrt{2} } } } \cdot \ldots}\)

Eariu52

\(\displaystyle{ \frac{\left( 7-k\right) \pi }{12} = \sum_{n=0}^{ \infty } \frac{ \sqrt{3} }{12n+1} - \frac{k}{12n+3} + \frac{2}{12n+5} - \frac{2}{12n+7} + \frac{k}{12n+9} - \frac{ \sqrt{3} }{12n+11}}\)

Eariu52

\(\displaystyle{ \frac{\left( 3k+2\right) \pi }{12} = \sum_{n=0}^{ \infty } \frac{k \sqrt{3} }{12n+1} - \frac{\left( 2k-1\right) \sqrt{3}}{12n+3} + \frac{1}{12n+5} - \frac{1}{12n+7} + \frac{\left( 2k-1\right) \sqrt{3}}{12n+9} - \frac{k \sqrt{3} }{12n+11} , k \ge 1 }\)

Eariu52

\(\displaystyle{ \frac{\left( k+2\right) \pi }{6} = \sum_{n=0}^{ \infty } \frac{k+1}{12n+1} - \frac{k \sqrt{3} }{12n+3} + \frac{1}{12n+5} - \frac{1}{12n+7} + \frac{k \sqrt{3} }{12n+9} - \frac{k+1}{12n+11}, k \ge 1}\)

Eariu52

\(\displaystyle{ \frac{2 \pi }{3} = \sum_{n=0}^{ \infty } \frac{3 }{12n+1} - \frac{ 2\sqrt{3} }{12n+3} + \frac{1}{12n+5} - \frac{1}{12n+7} + \frac{ 2\sqrt{3} }{12n+9} - \frac{ 3 }{12n+11} }\)

Eariu52

\(\displaystyle{ \frac{5 \pi }{12} = \sum_{n=0}^{ \infty } \frac{ \sqrt{3} }{12n+1} - \frac{ \sqrt{3} }{12n+3} + \frac{1}{12n+5} - \frac{1}{12n+7} + \frac{ \sqrt{3} }{12n+9} - \frac{ \sqrt{3} }{12n+11} }\)

Eariu52

\(\displaystyle{ \sum_{n=2}^{ \infty } \frac{n+\left( -1\right) ^{n} }{n ^{4}-n ^{2} } = \frac{ \pi ^{2}-6 }{12} }\)

Eariu52

\(\displaystyle{ \frac{2 \sqrt{2} }{3}=\left( \sqrt{2+ \sqrt{2} } - \sqrt{2- \sqrt{2} }\right) \cdot \left( \sqrt{2+ \sqrt{2+ \sqrt{2} } } - \sqrt{2- \sqrt{2- \sqrt{2} } }\right)\cdot \ldots }\)

Eariu52

\(\displaystyle{ \lim_{ n\to \infty } \sum_{k=1}^{n} \left( \frac{k}{2n-k} \right) ^{k} = \frac{e ^{2} }{e ^{2}-1 } }\)

Eariu52

\(\displaystyle{ \prod_{n=1}^{ \infty } \frac{1 + \sum_{k=1}^{n}a ^{k} }{\sum_{k=1}^{n}a ^{k} } = \frac{a}{a-1}, a>1 }\)

Eariu52

\(\displaystyle{ S _{1}= \sum_{n=1}^{ \infty } \frac{1}{ \prod_{k=1}^{2n-1} \left( k+1- \frac{1}{k+1} \right) } }\)

\(\displaystyle{ S _{2}= \sum_{n=1}^{ \infty } \frac{1}{ \prod_{k=1}^{2n} \left( k+1- \frac{1}{k+1} \right) } }\)

\(\displaystyle{ \frac{S _{2} }{S _{1} }= \frac{e}{2} - 1 }\)

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