Nieznany wzór na znaną liczbę
Nieznany wzór na znaną liczbę
\(\displaystyle{ \pi = 6-\sum_{n=1}^{ \infty } \frac{1}{\left( n+ \frac{1}{2} \right) \left( n- \frac{1}{2} \right) } + \frac{1}{2\left( n+ \frac{1}{4} \right)\left( n- \frac{1}{4} \right) } }\)
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Re: Nieznany wzór na znaną liczbę
\(\displaystyle{ A= \sum_{n=2}^{ \infty } \frac{1}{n ^{2}-1 } = \frac{3}{4}}\)
\(\displaystyle{ B= \frac{1}{1} \cdot \frac{1}{3} + \frac{1}{5} \cdot \frac{1}{7}+ \frac{1}{9} \cdot \frac{1}{11} +...= \frac{ \pi }{8} }\)
\(\displaystyle{ B= \frac{1}{2 ^{2}-1 } + \frac{1}{6 ^{2}-1 } + \frac{1}{10 ^{2}-1 } +...}\)
\(\displaystyle{ \frac{6- \pi }{8} =A-B}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{\left( 4n-1\right) ^{2}-1} + \frac{1}{\left( 4n\right) ^{2}-1 } + \frac{1}{\left( 4n+1\right)^{2}-1 }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{\left( 4n-1\right) ^{2}-1} + \frac{1}{\left( 4n+1\right)^{2}-1 } + \frac{1}{\left( 4n\right) ^{2}-1 }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{16n ^{2}+1-8n-1 } + \frac{1}{16n ^{2} +1+8n-1} + \frac{1}{\left( 4n+1\right)\left( 4n-1\right) } }\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{32n ^{2} }{256n ^{4}-64n ^{2} } + \frac{1}{4\left( n+ \frac{1}{4} \right)4\left( n- \frac{1}{4} \right) } }\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8n ^{2}-2} + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8\left( n ^{2} - \frac{1}{4} \right) } + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8\left( n+ \frac{1}{2} \right)\left( n- \frac{1}{2} \right) } + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)
\(\displaystyle{ B= \frac{1}{1} \cdot \frac{1}{3} + \frac{1}{5} \cdot \frac{1}{7}+ \frac{1}{9} \cdot \frac{1}{11} +...= \frac{ \pi }{8} }\)
\(\displaystyle{ B= \frac{1}{2 ^{2}-1 } + \frac{1}{6 ^{2}-1 } + \frac{1}{10 ^{2}-1 } +...}\)
\(\displaystyle{ \frac{6- \pi }{8} =A-B}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{\left( 4n-1\right) ^{2}-1} + \frac{1}{\left( 4n\right) ^{2}-1 } + \frac{1}{\left( 4n+1\right)^{2}-1 }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{\left( 4n-1\right) ^{2}-1} + \frac{1}{\left( 4n+1\right)^{2}-1 } + \frac{1}{\left( 4n\right) ^{2}-1 }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{16n ^{2}+1-8n-1 } + \frac{1}{16n ^{2} +1+8n-1} + \frac{1}{\left( 4n+1\right)\left( 4n-1\right) } }\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{32n ^{2} }{256n ^{4}-64n ^{2} } + \frac{1}{4\left( n+ \frac{1}{4} \right)4\left( n- \frac{1}{4} \right) } }\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8n ^{2}-2} + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8\left( n ^{2} - \frac{1}{4} \right) } + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)
\(\displaystyle{ \frac{6- \pi }{8} = \sum_{n=1}^{ \infty } \frac{1}{8\left( n+ \frac{1}{2} \right)\left( n- \frac{1}{2} \right) } + \frac{1}{16\left( n+ \frac{1}{4} \right) \left( n- \frac{1}{4} \right) }}\)