円周率の近似
1 π = 2 2 99 2 ∑ n = 0 ∞ ( 4 n ) ! ( 1103 + 26390 n ) ( 4 n 99 n n ! ) 4 {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{n=0}^{\infty }{\frac {(4n)!(1103+26390n)}{(4^{n}99^{n}n!)^{4}}}}
4 π = ∑ n = 0 ∞ ( − 1 ) n ( 21460 n + 1123 ) ( 4 n ! ) 882 2 n + 1 ( 4 n n ! ) 4 {\displaystyle {\frac {4}{\pi }}=\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}(21460n+1123)(4n!)}{{882}^{2n+1}{(4^{n}n!)}^{4}}}}
π 4 = 2 arctan  1 3 + arctan  1 7 {\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}}
π 4 = arctan  1 2 + arctan  1 3 {\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}}
π 4 = 5 arctan  1 7 + 2 arctan  3 79 {\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}