设(1/2+2/3+3/4+...+2011/2012)为x,(1+1/2+2/3+...+2010/2011)为y
原式=xy-(x+1)(y-1)
=xy-y+1-xy+x
=x-y+1
=(1/2+2/3+3/4+...+2011/2012)-(1+1/2+2/3+...+2010/2011)+1
=2011/2012