1/1*3+1/3*5+.+1/2003*2005
=(1/2)*(1-1/3)+(1/2)*(1/3-1/5)+……+(1/2)*(1/2003-1/2005)
=(1/2)*(1-1/3+1/3-1/5+1/5-1/7+……+1/2003-1/2005)
=(1/2)*(1-1/2005)
=(1/2)*(2004/2005)
=1002/2005