1984 AMC 12 Question 30

For any complex number $w=a+bi,$ $|w|$ is defined to be the real number $\sqrt{a^2+b^2}.$ If $w=\cos40^\circ+i\sin40^\circ,$ then $$
|w+2w^2+3w^3+\cdots+9w^9|^{-1}
$$ equals