Best Formulas for Pi (Other Than Circumference Divided by Diameter)

The Top Ten

1 pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 ...

2 pi/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * ...

3 pi^2 / 6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...

4 pi = ln(i-root(-1)) (e^(i*pi) = -1)

5 sqrt(pi)/2 = .5!

6 pi/2 = sqrt(2)/2 * sqrt(2+sqrt(2))/2 * sqrt(2+sqrt(2+sqrt(2)))/2 ...

7 pi = 4/(1+(1^2/(2+(3^2/(2+(5^2/(2+(7^2/ ...

8 pi = (a(n+1) + b(n+1))^2 / 4(t(n+1)) given, a(0)=1, b(0)=1/sqrt(2), t(0)=1/4, p(0)=1, a(n+1)=(a(n)+b(n))/2, b(n+1)=sqrt(a(n)*b(n)), t(n+1)=t(n) - p(n)*(a(n)-a(n+1))^2, p(n+1)=2p(n)

9 pi = sigma n=0 to infinity (1/16^n) * (4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6)

10 1/pi = 2sqrt(2)/9801 * sigma n=0 to infinity ((4n)!*(1103+26390n)) / ((n!)^4*(396^(4n)))

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