=1+(1/3+1/6)+(1/10+1/15)+(1/21+1/28)+…+(1/(2009*1004)+1/(2009*2005))+1/(2011*1005)
=1+1/(1*2)+1/(2*3)+1/(3*4)+…+1/(1004*1005)+1/(2011*1005)
=1+1-1/2+1/2-1/3+1/3-1/4+…+1/1004-1/1005+1/(2011*1005)
=1+1-1/1005+1/(2011*1005)
=2-2011/(2011*1005)+1/(2011*1005)
=2-2010/(2011*1005)
=4020/2011