円周率を含む数式
円周率を含む級数
( x ) n {\displaystyle (x)_{n}} は階乗冪。
}}} , ただし B2n は ベルヌーイ数)
∑ n = 1 ∞ 3 n − 1 4 n ζ ( n + 1 ) = π {\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi }
∑ n = 0 ∞ ( ( − 1 ) n 2 n + 1 ) 1 = 1 1 − 1 3 + 1 5 − 1 7 + 1 9 − ⋯ = arctan  1 = π 4 {\displaystyle \sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}} (ライプニッツ公式)
∑ n = 1 ∞ ( − 1 ) n + 1 n 2 = 1 1 2 − 1 2 2 + 1 3 2 − 1 4 2 + ⋯ = π 2 12 {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{12}}}
C = π d {\displaystyle C=\pi d}
( 1657145277365 + 212175710912 61 + ( 107578229802750 + 3773980892672 61 ) n ) ( 3 n ) ! ( n ! ) 3 ( 1249638720 + 159999840 61 ) n = 1 π {\displaystyle {\frac {12}{\sqrt {(1249638720+159999840{\sqrt {61}})^{3}}}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}(6n)!(1657145277365+212175710912{\sqrt {61}}+(107578229802750+3773980892672{\sqrt {61}})n)}{(3n)!(n!)^{3}(1249638720+159999840{\sqrt {61}})^{n}}}={\frac {1}{\pi }}} (Borwein)
3 6 5 ∑ k = 0 ∞ ( ( 4 k ) ! ) 2 ( 6 k ) ! 9 k + 1 ( 12 k ) ! ( 2 k ) ! ( 127169 12 k + 1 − 1070 12 k + 5 − 131 12 k + 7 + 2 12 k + 11 ) = π {\displaystyle {\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!