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\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 π