X/(1×2)+X/(2×3)+X/(3×4)+.+X/(2003×2004) = 2003X×{ 1/(1×2)+1/(2×3)+1/(3×4)+.+1/(2003×2004) } = 2003X×{ 1-1/2+1/2-1/3+1/3-1/4+.+1/2003-1/2004 } = 2003X×{ 1-1/2004 } = 2003X×2003/2004= 2003...