对于正数x,规定f(x)=x/(1+x),例如f(3)=3/(1+3)=3/4,f(1/3)=(1/3)/(1+1/3)=1/4.计算:f(1/2009)+f(1/2008)+…+f(1/2)+f(1)+f(1)+f(2)+…+f(2008)+f(2009)计算:f(1/2014)+f(1/2013)+…+f(1/2)+f(1)+f(1)+f(2)+…+f(2013)+f(2014)

Question

对于正数x,规定f(x)=x/(1+x),
例如f(3)=3/(1+3)=3/4,f(1/3)=(1/3)/(1+1/3)=1/4.计算:f(1/2009)+f(1/2008)+…+f(1/2)+f(1)+f(1)+f(2)+…+f(2008)+f(2009)
计算:f(1/2014)+f(1/2013)+…+f(1/2)+f(1)+f(1)+f(2)+…+f(2013)+f(2014)

Answer 1

由f(x) = x / (1+x)容易证得对任意x≠0,-1,f(x) + f(1 / x) = 1.于是f(1/2014) + f(1/2013) + … + f(1/2) + f(1) + f(1) + f(2) + … + f(2013) + f(2014).=(f(1/2014) + f(2014)) + (f(1/2013) + f(2013)) + ...+ ...