(1/2)((1-1/2)+(1/4-1/6)+(1/6-1/8)+...+(1/2006-1/2008))=(1/2)(1-1/2+1/4-1/2008)=3/8-1/4016=1505/4016
1/1×2+1/4×6+1/6×8+···+1/2006×2008=1/2+1/2{1/4-1/6+1/6-1/8+......+1/2006-1/2008 }=1/2+1/2{1/4-1/2008}=2509/4016
