Hello, After reading some articles on the Brent-Salamin algorithm based on the AGM and algorithms of the Borweins brothers arising from the famous Ramanujan formulas on the Pi number. I wondered if it was possible to find an algorithm for any convergence speed m considered. Finally I came to establish the following general formula which is of convergence (2 * m): x(k+1)=x(k)+sum(2*((m!)^2)*sin(n*x(k))/(n*((m+n)!)*((m-n)!)),n=1..m); x(0) = 3 ; k --- infinity ; x(k) ---- Pi ; If we take m = 5, the convergence is (2 * m) = 10; we are getting: x(k+1)= x(k)+(5/3)*sin(x(k))+(10/21)*sin(2*x(k))+(5/42)*sin(3*x(k))+(5/252)*sin(4*x(k))+(1/630)*sin(5*x(k)); x(0)=3; x(1)=3.141592653589628958434997482316410876703079409530919198836399516902375075829274562410138150533960715 ; x(2)=3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 ; So if I want to write a 2000000 convergence formula, then I would simply have: x(k+1)=x(k)+sum(2*((1000000!)^2)*sin(n*x(k))/(n*((1000000+n)!)*((1000000-n)!)),n=1..1000000); Unfortunately, this formula can not be useful for the calculation of Pi given the difficulty related to the calculation of sinuses. Best regards...