1 / 1 * 2 + 1 / 2 * 3 + 1 / 3 * 4 + ·· + 1 / 1000 * 1001

1 / 1 * 2 + 1 / 2 * 3 + 1 / 3 * 4 + ·· + 1 / 1000 * 1001

1/(1×2)+1/(2×3)+1/(3×4)+...+1/(1000×1001)
=1-1/2+1/2-1/3+1/3-1/4+...+1/1000-1/1001
=1-1/1001
=1000/1001