[math-fun] some formulas of pi in infinte product (2)
Hello, The first writing of my previous formulas was not very clear .. Their writing is more appropriate in introducing infinite limits. We will have: Pi=Limit(2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((2-1/(cos(Pi*2^(-i-2)))^2)^(2^(i-1)),i=1..n))^(1/2^n))),n=infinity); Pi=Limit(2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product((1-tan(Pi*2^(-i-2))^2)^(2^(i-1)),i=1..n))^(1/2^n))),n=infinity); Pi=2^(n+5/2)*sqrt(1-1/(2^(2^(-n-1))*(product(2^(2^(i-1))/(1/cos(Pi*2^(-i-1))+1)^(2^(i-1)),i=1..n))^(1/2^n))); Pi/4=Limit(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^(-i-2))^2)^(2^(i-1)),i=1..n))^(1/2^n)+2^(-4*n-2))),n=infinity); Pi/4=Limit(sqrt(3*2^(2*n+1)-sqrt(3)*2^(4*n+2)*sqrt(2^(-2^(-n-1)-4*n-1)/(product((2-1/(cos(Pi*2^(-i-2)))^2)^(2^(i-1)),i=1..n))^(1/2^n)+2^(-4*n-2))),n=infinity); Best regards.
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François Mendzina Essomba