1/3*5+1/5*7+1/7*9+.+1/2001*2003=1/2[1/3-1/5]+1/2[1/5-1/7]+1/2[1/7-1/9]+...+1/2[1/2001-1/2003]=1/2[1/3-1/5+1/5-1/7+1/7-1/9+...+1/2001-1/2003]=1/2[1/3-1/2003]=1000/6009