原式=2005*(1/(2*3)+1/(3*4)+...+1/(2003*2004))+1/2004
=2005*(1/2-1/3+1/3-1/4+...+1/2003-1/2004)+1/2004
=2005*(1/2-1/2004)+1/2004
=2005/2-1
=2003/2