Hello, Formula 40 Limit(2^(n)*((product(x(i),i=1..m))^(1/2^n)-1)/((product(x(i),i=1..m))^(1/(2^(n+1)))),n=infinity)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity),i=1..m); Formula 41 Limit(2^(n)*((product(x(i)/y(i),i=1..m))^(1/2^n)-1)/((product(x(i)/y(i),i=1..m))^(1/(2^(n+1)))),n=infinity,right)=sum(Limit(2^(n)*(x(i)^(1/2^n)-1)/(x(i)^(1/(2^(n+1)))),n=infinity),i=1..m)-sum(Limit(2^(n)*(y(i)^(1/2^n)-1)/(y(i)^(1/(2^(n+1)))),n=infinity),i=1..m); Formula 42 Pi=Limit(2^(n)*(exp(2^(-n)*Pi)-1)/exp(2^(-(n+1)*Pi)),n=infinity); Formula 43 cosh(x)=Limit(2^(n-1)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))),n=infinity); formula 44 sinh(x)=Limit(2^(n-1)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))),n=infinity); formula 45 tanh(x)=Limit(((1/2)*2^(n)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))))/((1/2)*2^(n)*((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x))))),n=infinity); Formula 46 cotanh(x)= (((exp(1/(2^(n)*exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))+(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x)))))/((1/2)*2^(n)*((exp(x)))-1)/exp(1/(2^(n+1)*exp(x)))-(exp(1/(2^(n)*exp(-x)))-1)/exp(1/(2^(n+1)*exp(-x))))); Formula 47 cos(x)=Limit(2^n*exp(cos(x)/2^n-2^(-n-1)*cos(x))*cos(2^(-n-1)*sin(x))-2^n*exp(-2^(-n-1)*cos(x))*cos(2^(-n-1)*sin(x)),n=infinity); Formula 48 sin(x)=Limit(2^n*exp(cos(x)/2^n-2^(-n-1)*cos(x))*sin(2^(-n-1)*sin(x))+2^n*exp(-2^(-n-1)*cos(x))*sin(2^(-n-1)*sin(x)),n=infinity); Formula 49 Limit(-2^(2*n+1)*cos(sin(x)/2^n)+2^(2*n)*exp(2^(1-n)*cos(x)-cos(x)/2^n)+2^(2*n)*exp(-cos(x)/2^n),n=infinity)= 1; Best regards