פורטל:מתמטיקה/נוסחה נבחרת/16

• מגוון נוסחאות ל-$\pi$
  • $${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots$$
  • $$\pi ={\frac {3}{4}}{\sqrt {3}}+24\left({\frac {1}{12}}-\sum _{n=2}^{\infty }{\frac {(2n-2)!}{(n-1)!^{2}(2n-3)(2n+1)2^{4n-2}}}\right)$$
  • $\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}$
  • ${\frac {1}{\pi }}=12\sum _{q=0}^{\infty }{\frac {(-1)^{q}(6q)!(545140134q+13591409)}{(3q)!(q!)^{3}\left(640320\right)^{3q+3/2}}}$
  • $\zeta (2):=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}$
  • $${\frac {\pi }{4}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots$$
  • ${\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,$
  • $\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdot {\frac {64}{63}}\cdots ={\frac {\pi }{2}}$ (see also Wallis product)
  • $\left({\frac {1}{2}}\right)!:=\Gamma \left({\frac {3}{2}}\right):=\int _{0}^{\infty }{\sqrt {x}}e^{-x}\,dx={\frac {\sqrt {\pi }}{2}}$