1/2*4+1/4*6+1/6*8+.+1/2010*2012=(1/4)*(1/1*2+1/2*3+1/3*4+...+1/1005*1006)=(1/4)*[(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/1005-1/1006)]=(1/4)*(1-1/1006)=1005/4024