Hello... I noticed that : atan(x)=(205*x^7+2184*x^5+5760*x^3+4096*x)/(36*x^8+820*x^6+4368*x^4+7680*x^2+4096)+sum(S(n),n=1..infinity); with S(m)=2^(-m-2)*x*sum(1/((2*k-1)^2*2^(-2*m-4)*x^2+1)-1/(k^2*2^(-2*m-2)*x^2+1),k,1,2^(m+1)); we deduce from it: 1) Pi= 2449/850 + 4*sum(S(n),n=1..infinity); with S(m)=2^(-m-2)*sum(1/((2*k-1)^2*2^(-2*m-4)+1)-1/(k^2*2^(-2*m-2)+1),k,1,2^(m+1)) ; 2) Pi=139701*sqrt(3)/91504 + 3*sum(S(n),n=1..infinity); with S(m)=sqrt(3)*2^(-m-2)*sum(1/(3*(2*k-1)^2*2^(-2*m-4)+1)-1/(3*k^2*2^(-2*m-2)+1),k,1,2^(m+1)) ; 3) Pi=169189*sqrt(3)/96824 + 6*sum(S(n),n=1..infinity); with (2^(-m-2)*sum(1/(((2*k-1)^2*2^(-2*m-4))/3+1)-1/((k^2*2^(-2*m-2))/3+1),k,1,2^(m+1)))/sqrt(3);