円周率を含む数式
A = π r 2 {\displaystyle A=\pi r^{2}}
∑ k = 0 ∞ k ! ( 2 k + 1 ) ! ! = ∑ k = 0 ∞ 2 k k ! 2 ( 2 k + 1 ) ! = π 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}}
12 ∑ k = 0 ∞ ( − 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 = 1 π {\displaystyle 12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}={\frac {1}{\pi }}} (Chudnovsky algorithm)
2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k = 1 π {\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {1}{\pi }}} (シュリニヴァーサ・ラマヌジャン)
12 ( 1249638720 + 159999840 61 ) 3 ∑ n = 0 ∞ ( − 1 ) n ( 6 n ) !
π 4 = 3 4 ⋅ 5 4 ⋅ 7 8 ⋅ 11 12 ⋅ 13 12 ⋅ 17 16 ⋅ 19 20 ⋅ 23 24 ⋅ 29 28 ⋅ 31 32 ⋅ ⋯ {\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{16}}\cdot {\frac {19}{20}}\cdot {\frac {23}{24}}\cdot {\frac {29}{28}}\cdot {\frac {31}{32}}\cdot \,\cdots } (オイラー)
全ての奇素数を分子とし、それに最も近い4の倍数を分母とした分数の総乗。