1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+---+1/(a+2004)-1/(a+2005)
=1/a-1/(a+2005)
=2005/(a²+2005a)