原式=1/2-1/3+1/3-1/4+1/4-1/5+…+1/2003-1/2004
=1/2+(-1/3+1/3)+(-1/4+1/4)+…+(-1/2003+1/2003)-1/2004
=1/2-1/2004
=(1002-1)/2004
=1001/2004