π
{\displaystyle \pi }
bıvurne
Nilakantha Somayaji :
π
=
3
+
4
3
3
−
3
−
4
5
3
−
5
+
4
7
3
−
7
−
4
9
3
−
9
+
.
.
.
{\displaystyle \pi ={3}+{\frac {4}{{3^{3}}-3}}-{\frac {4}{{5^{3}}-5}}+{\frac {4}{{7^{3}}-7}}-{\frac {4}{{9^{3}}-9}}+...}
π
=
3
+
4
2
×
3
×
4
−
4
4
×
5
×
6
+
4
6
×
7
×
8
−
4
8
×
9
×
10
+
.
.
.
{\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+...}
Franciscus Vieta :
π
=
2
×
2
2
×
2
2
+
2
×
2
2
+
2
+
2
×
2
2
+
2
+
2
+
2
×
⋯
{\displaystyle \pi =2\times {\frac {2}{\sqrt {2}}}\times {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}\times \cdots }
Gregory–Leibniz :
π
=
4
∑
n
=
0
∞
(
−
1
)
n
2
n
+
1
=
4
(
1
1
−
1
3
+
1
5
−
1
7
+
−
⋯
)
=
4
1
+
1
2
2
+
3
2
2
+
5
2
2
+
⋱
{\displaystyle \pi =4\sum _{n=0}^{\infty }{\cfrac {(-1)^{n}}{2n+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+-\cdots \right)\!={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}\!}
Isaac Newton :
π
=
∑
n
=
0
∞
2
n
+
1
⋅
(
n
!
)
2
(
2
n
+
1
)
!
{\displaystyle \pi =\sum _{n=0}^{\infty }{\cfrac {{2^{n+1}}\cdot {(n!)^{2}}}{(2n+1)!}}}
Leonhard Euler :
π
=
−
i
ln
(
−
1
)
{\displaystyle \pi =-i\ln(-1)}
Bailey -Borwein-Plouffe :
π
=
∑
n
=
0
∞
(
1
16
)
n
(
4
8
n
+
1
−
2
8
n
+
4
−
1
8
n
+
5
−
1
8
n
+
6
)
{\displaystyle \pi =\sum _{n=0}^{\infty }{\biggl (}{\frac {1}{16}}{\biggr )}^{n}\left({\frac {4}{8n+1}}-{\frac {2}{8n+4}}-{\frac {1}{8n+5}}-{\frac {1}{8n+6}}\right)}
