1-1/2-1/6-1/12-1/20-...-1/2011*2012=1-(1/1*2+1/2*3+1/3*4+1/4*5+...+1/2011*2012)=1-[(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/2011-1/2012)]=1-(1-1/2012)=1/2012