円周率の近似
π 4 = 4 arctan  1 5 − arctan  1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}
π 4 = 1 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + 1 13 ⋯ {\displaystyle {\frac {\pi }{4}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}+{\frac {1}{13}}\cdots }
π 4 = 12 arctan  1 49 + 32 arctan  1 57 − 5 arctan  1 239 + 12 arctan  1 110443 {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}
2 π = 1 2 1 2 + 1 2 1 2 + 1 2 + 1 2 ⋯ {\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}{\sqrt {{\frac {1}{2}}+{\sqrt {\frac {1}{2}}}}}{\sqrt {{\frac {1}{2}}+{\sqrt {{\frac {1}{2}}+{\sqrt {\frac {1}{2}}}}}}}\cdots }
π = 4 1 + 1 2 3 + 2 2 5 + ⋯ ⋯ + n 2 ( 2 m + 1 ) + ⋯ {\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {\cdots }{\cdots +{\cfrac {n^{2}}{(2m+1)+\cdots }}}}}}}}}}}