[math-fun] Back to square one (Archimedes type formula)

An omission of limits in two previous formulas 1/sqrt(Pi)=Limit((8/(m*(Gamma(1/4))^2))*sum((1-(k/m)^4)^(1/4),k=1..m-1),m=infinity); (Pi)^(3/2)=Limit((4*((Gamma(3/4))^2/(m)))*sum((1-(k/m)^4)^(1/4),k=1..m-1),m=infinity); Back to square one Archimedes type formula a_(n+1):= 2^(n+3/2)*(3-(3+(3*(3-a_(n)^2/(2^(2*n+1)))^2+9)^(1/2))^(1/2))^(1/2); a_(1):= 2*(6-2*3^(1/2))^(1/2); n---infinity; a_(n)--- Pi We obtain by developing this formula: Pi=2^(n+3/2)*sqrt(3-sqrt(3+3*sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+_)))))))); we have n (*) times the number (2); For example: Pi~2^(6+3/2)*sqrt(3-sqrt(3+3*sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2)))))))); I suspect that there are an infinity of his formulas FME...