1/(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)……1/(x+2008)(x+2009)
=1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3).+1/(x+2008)-1/(x+2009)
=2/(x+1)-1/(x+2009)
1/(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)……1/(x+2008)(x+2009)
=1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3).+1/(x+2008)-1/(x+2009)
=2/(x+1)-1/(x+2009)