1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+···+1/（x+2010+（x+2011）

原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+……+1/(x+2010)-1/(x+2011)
=1/x-1/(x+2011)
=2011/x(x+2011)