\begin{aligned} \frac{1}{\pi} &=\frac{1}{8} \sum^{\infty}_{m=0}(20m+3)\frac{(-1)^m(4m)!}{(4\sqrt{2})^{4m}(m!)^4} & \quad \frac{1}{\pi} &=\frac{1}{2\sqrt{3}} \sum^{\infty}_{m=0}(8m+1)\frac{(4m)!}{(4\sqrt{3})^{4m}(m!)^4} \\ \frac{1}{\pi} &=\frac{\sqrt3}{16} \sum^{\infty}_{m=0}(28m+3)\frac{(-1)^m(4m)!}{(64\sqrt{3})^{2m}(m!)^4} & \quad \frac{1}{\pi} &=\frac{2\sqrt{2}}{9} \sum^{\infty}_{m=0}(10m+1)\frac{(4m)!}{12^{4m}(m!)^4} \\ \frac{1}{\pi} &=\frac{1}{72} \sum^{\infty}_{m=0}(260m+23)\frac{(-1)^m(4m)!}{(12\sqrt{2})^{4m}(m!)^4} & \quad \frac{1}{\pi} &=\frac{3\sqrt3}{49} \sum^{\infty}_{m=0}(40m+3)\frac{(4m)!}{28^{4m}(m!)^4} \\ \frac{1}{\pi} &=\frac{1}{18\sqrt{11}} \sum^{\infty}_{m=0}(280m+19)\frac{(4m)!}{(12\sqrt{11})^{4m}(m!)^4} & \quad \frac{1}{\pi} &=\frac{\sqrt{5}}{288} \sum^{\infty}_{m=0}(644m+41)\frac{(-1)^m(4m)!}{(1152\sqrt{5})^{2m}(m!)^4} \\ \frac{1}{\pi} &=\frac{2}{84^2} \sum^{\infty}_{m=0}(21460m+1123)\frac{(-1)^m(4m)!}{(84\sqrt{2})^{4m}(m!)^4} & \quad \frac{1}{\pi} &=\frac{2\sqrt{2}}{99^2} \sum^{\infty}_{m=0}(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^4} \end{aligned}
π
1
π
1
π
1
π
1
π
1
=
8
1
m
=
0
∑
∞
(
2
0
m
+
3
)
(
4
2
)
4
m
(
m
!
)
4
(
−
1
)
m
(
4
m
)
!
=
1
6
3
m
=
0
∑
∞
(
2
8
m
+
3
)
(
6
4
3
)
2
m
(
m
!
)
4
(
−
1
)
m
(
4
m
)
!
=
7
2
1
m
=
0
∑
∞
(
2
6
0
m
+
2
3
)
(
1
2
2
)
4
m
(
m
!
)
4
(
−
1
)
m
(
4
m
)
!
=
1
8
1
1
1
m
=
0
∑
∞
(
2
8
0
m
+
1
9
)
(
1
2
1
1
)
4
m
(
m
!
)
4
(
4
m
)
!
=
8
4
2
2
m
=
0
∑
∞
(
2
1
4
6
0
m
+
1
1
2
3
)
(
8
4
2
)
4
m
(
m
!
)
4
(
−
1
)
m
(
4
m
)
!
π
1
π
1
π
1