Trying to understand the formula of FME Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^ (-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n))); where the first occurrence of n does not seem to have a value and is not part of a product. (?) 2017-10-31 4:26 GMT-06:00 François Mendzina Essomba via math-fun < math-fun@mailman.xmission.com>:
Hello,
Some formulas of pi in infinite product:
(4/Pi)=product(1/(1-tan(Pi*2^(-n-3))^2),n=0..infinity);
(4/Pi)=product((1/2)*(1+1/cos(Pi*2^(-n-2))),n=0..infinity);
Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^ (-n-2)))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)));
Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((1-tan(Pi*2^(-n -2))^2)^(2^(n-1)),n=1..infinity))^(1/2^n)));
Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product(2^(2^(n-1))/(1/ cos(Pi*2^(-n-1))+1)^(2^(n-1)),n=1..infinity))^(1/2^n)));
Pi/4=sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4* n-1)/(product((1-tan(Pi*2^(-n-2))^2)^(2^(n-1)),n=1..infinity))^(1/ 2^n)+2^(-4*n <https://maps.google.com/?q=2%5En)%2B2%5E(-4*n&entry=gmail&source=g>-2)));
Pi/4=sqrt(3*2^(2*n+1 <https://maps.google.com/?q=2*n%2B1&entry=gmail&source=g>)-sqrt(3)*2^(4*n <https://maps.google.com/?q=2%5E(4*n&entry=gmail&source=g> +2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((2-1/(cos(Pi*2^(-n-2)))^2)^( 2^(n-1)),n=1 <https://maps.google.com/?q=2%5E(n-1)),n%3D1&entry=gmail&source=g>..infi nity))^(1/2^n)+2^(-4*n <https://maps.google.com/?q=2%5En)%2B2%5E(-4*n&entry=gmail&source=g>-2)));
4/Pi=product(((cos((2*Pi)/8^n)+cos(Pi/8^n))*sec((2*Pi)/8^n)) /8+(cos((3*Pi)/(2*8^n))*sec((2*Pi)/8^n))/4+((cos(Pi/8^n)+1)* sec((2*Pi)/8^n))/8+(cos(Pi/(2*8^n))*sec((2*Pi)/8^n))/4,n=1..infinity);
Best regards.
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