List of formulae involving π
Updated: 11/6/2025, 2:11:03 AM Wikipedia source
The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π.
Tables
· Formulae yielding <i>π</i> › Other infinite series
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
4
n
)
!
(
21460
n
+
1123
)
(
n
!
)
4
441
2
n
+
1
2
10
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}^{10n+1}}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
6
n
+
1
)
(
1
2
)
n
3
4
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(6n+1)\left({\frac {1}{2}}\right)_{n}^{3}}{{4^{n}}(n!)^{3}}}}
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
32
Z
{\displaystyle \pi ={\frac {32}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
5
−
1
2
)
8
n
(
42
n
5
+
30
n
+
5
5
−
1
)
(
1
2
)
n
3
64
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {{\sqrt {5}}-1}{2}}\right)^{8n}{\frac {(42n{\sqrt {5}}+30n+5{\sqrt {5}}-1)\left({\frac {1}{2}}\right)_{n}^{3}}{{64^{n}}(n!)^{3}}}}
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
27
4
Z
{\displaystyle \pi ={\frac {27}{4Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
2
27
)
n
(
15
n
+
2
)
(
1
2
)
n
(
1
3
)
n
(
2
3
)
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {2}{27}}\right)^{n}{\frac {(15n+2)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}}
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
15
3
2
Z
{\displaystyle \pi ={\frac {15{\sqrt {3}}}{2Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
4
125
)
n
(
33
n
+
4
)
(
1
2
)
n
(
1
3
)
n
(
2
3
)
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(33n+4)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}}
π
=
85
85
18
3
Z
{\displaystyle \pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}}
π
=
85
85
18
3
Z
{\displaystyle \pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
85
85
18
3
Z
{\displaystyle \pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
4
85
)
n
(
133
n
+
8
)
(
1
2
)
n
(
1
6
)
n
(
5
6
)
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{85}}\right)^{n}{\frac {(133n+8)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}}
π
=
5
5
2
3
Z
{\displaystyle \pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}}
π
=
5
5
2
3
Z
{\displaystyle \pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
5
5
2
3
Z
{\displaystyle \pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
4
125
)
n
(
11
n
+
1
)
(
1
2
)
n
(
1
6
)
n
(
5
6
)
n
(
n
!
)
3
{\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(11n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}}
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
3
Z
{\displaystyle \pi ={\frac {2{\sqrt {3}}}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
8
n
+
1
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
9
n
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(8n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{n}}}}
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
3
9
Z
{\displaystyle \pi ={\frac {\sqrt {3}}{9Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
40
n
+
3
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
49
2
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(40n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{49}^{2n+1}}}}
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
11
11
Z
{\displaystyle \pi ={\frac {2{\sqrt {11}}}{11Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
280
n
+
19
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
99
2
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(280n+19)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{99}^{2n+1}}}}
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
2
4
Z
{\displaystyle \pi ={\frac {\sqrt {2}}{4Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
10
n
+
1
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
9
2
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(10n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{2n+1}}}}
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
5
5
Z
{\displaystyle \pi ={\frac {4{\sqrt {5}}}{5Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
644
n
+
41
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
5
n
72
2
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(644n+41)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}}
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
3
3
Z
{\displaystyle \pi ={\frac {4{\sqrt {3}}}{3Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
28
n
+
3
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
3
n
4
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
4
Z
{\displaystyle \pi ={\frac {4}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
20
n
+
3
)
(
1
2
)
n
(
1
4
)
n
(
3
4
)
n
(
n
!
)
3
2
2
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{2}^{2n+1}}}}
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
72
Z
{\displaystyle \pi ={\frac {72}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
4
n
)
!
(
260
n
+
23
)
(
n
!
)
4
4
4
n
18
2
n
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}}}
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
π
=
1
Z
{\displaystyle \pi ={\frac {1}{Z}}}
π
=
3528
Z
{\displaystyle \pi ={\frac {3528}{Z}}}
Z
=
∑
n
=
0
∞
(
(
2
n
)
!
)
3
(
42
n
+
5
)
(
n
!
)
6
16
3
n
+
1
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}}
Z
=
∑
n
=
0
∞
(
−
1
)
n
(
4
n
)
!
(
21460
n
+
1123
)
(
n
!
)
4
4
4
n
882
2
n
{\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}}}
| π = 1 {Z}}} | Z = ∑ n = 0 ∞ ( ( 2 n ) ! ) 3 ( 42 n + 5 ) ( n ! ) 6 16 3 ^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}} |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( 4 n ) ! ( 21460 n + 1123 ) ( n ! ) 4 441 2 ^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}^{10n+1}}}} |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( 6 n + 1 ) ( 1 2 ) n 3 4 n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }{\frac {(6n+1)\left({\frac {1}{2}} ight)_{n}^{3}}{{4^{n}}(n!)^{3}}}} |
| π = 32 {Z}}} | Z = ∑ n = 0 ∞ ( 5 − 1 2 ) 8 n ( 42 n 5 + 30 n + 5 5 − 1 ) ( 1 2 ) n 3 64 n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {{\sqrt {5}}-1}{2}} ight)^{8n}{\frac {(42n{\sqrt {5}}+30n+5{\sqrt {5}}-1)\left({\frac {1}{2}} ight)_{n}^{3}}{{64^{n}}(n!)^{3}}}} |
| π = 27 4 {4Z}}} | Z = ∑ n = 0 ∞ ( 2 27 ) n ( 15 n + 2 ) ( 1 2 ) n ( 1 3 ) n ( 2 3 ) n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {2}{27}} ight)^{n}{\frac {(15n+2)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{3}} ight)_{n}\left({\frac {2}{3}} ight)_{n}}{(n!)^{3}}}} |
| π = 15 3 2 }}{2Z}}} | Z = ∑ n = 0 ∞ ( 4 125 ) n ( 33 n + 4 ) ( 1 2 ) n ( 1 3 ) n ( 2 3 ) n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}} ight)^{n}{\frac {(33n+4)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{3}} ight)_{n}\left({\frac {2}{3}} ight)_{n}}{(n!)^{3}}}} |
| π = 85 85 18 3 }}{18{\sqrt {3}}Z}}} | Z = ∑ n = 0 ∞ ( 4 85 ) n ( 133 n + 8 ) ( 1 2 ) n ( 1 6 ) n ( 5 6 ) n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{85}} ight)^{n}{\frac {(133n+8)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{6}} ight)_{n}\left({\frac {5}{6}} ight)_{n}}{(n!)^{3}}}} |
| π = 5 5 2 3 }}{2{\sqrt {3}}Z}}} | Z = ∑ n = 0 ∞ ( 4 125 ) n ( 11 n + 1 ) ( 1 2 ) n ( 1 6 ) n ( 5 6 ) n ( n ! ) 3 {\displaystyle Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}} ight)^{n}{\frac {(11n+1)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{6}} ight)_{n}\left({\frac {5}{6}} ight)_{n}}{(n!)^{3}}}} |
| π = 2 3 }}{Z}}} | Z = ∑ n = 0 ∞ ( 8 n + 1 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 9 ^{\infty }{\frac {(8n+1)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{9}^{n}}}} |
| π = 3 9 }{9Z}}} | Z = ∑ n = 0 ∞ ( 40 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 49 2 ^{\infty }{\frac {(40n+3)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{49}^{2n+1}}}} |
| π = 2 11 11 }}{11Z}}} | Z = ∑ n = 0 ∞ ( 280 n + 19 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 99 2 ^{\infty }{\frac {(280n+19)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{99}^{2n+1}}}} |
| π = 2 4 }{4Z}}} | Z = ∑ n = 0 ∞ ( 10 n + 1 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 9 2 ^{\infty }{\frac {(10n+1)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{9}^{2n+1}}}} |
| π = 4 5 5 }}{5Z}}} | Z = ∑ n = 0 ∞ ( 644 n + 41 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 5 ^{\infty }{\frac {(644n+41)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}} |
| π = 4 3 3 }}{3Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( 28 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 3 ^{\infty }{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}}} |
| π = 4 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( 20 n + 3 ) ( 1 2 ) n ( 1 4 ) n ( 3 4 ) n ( n ! ) 3 2 2 ^{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}} ight)_{n}\left({\frac {1}{4}} ight)_{n}\left({\frac {3}{4}} ight)_{n}}{(n!)^{3}{2}^{2n+1}}}} |
| π = 72 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( 4 n ) ! ( 260 n + 23 ) ( n ! ) 4 4 4 ^{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}}} |
| π = 3528 {Z}}} | Z = ∑ n = 0 ∞ ( − 1 ) n ( 4 n ) ! ( 21460 n + 1123 ) ( n ! ) 4 4 4 ^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}}} |
References
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- (integral form of arctan over its entire domain, giving the period of tan)
- The coefficients can be obtained by reversing the Puiseux series of z ↦
- The n {\displaystyle n} th root with the smallest positive principal argument
- When n ∈ Q +
- When n ∈ Q
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