Question

已知函数F(X)=1+X平方分之X平方,求F(1)+F(2)+F(二分之一)+F(3)+F(三分之一)+F(4)+F(四

Answer 1

首先证明：f(x)+f(1/x)=1

f(x)+f(1/x)=x^2/(1+x^2)+(1/x^2)/[1+(1/x^2)]

=x^2/(1+x^2)+x^2/(x^2+1)

=(1+x^2)/(x^2+1)=1

从而

f(1)+f(2)+f(1/2)+f(3)+f(1/3)+f(4)+f(1/4)

=f(1)+[f(2)+f(1/2)]+[f(3)+f(1/3)]+[f(4)+f(1/4)]

=1/(1+1)+1+1+1

=3+1/2

=7/2.