1/x(x+2)+1/(x+2)(x+4)+1/(x+4)(x+6)+……+1/(x+2000)(x+2002)
=1/2(1/x-1/(x+2)+1/(x+2)-1/(x+4)+...+1/(x+2000)-1/(x+2002)
=1/2[1/x-1/(x+2002)]
=1/2(x+2002-x)/x(x+2002)
=1001/x(x+2002)