Prove that the Taylor series about the origin of the function $[\log(1-z)]^2$ is given by $$\sum_{n=1}^{\infty} \frac{2H_{n}}{n+1} z^{n+1}$$ where $$H_{n} = \sum_{j=1}^{n}\frac{1}{j}$$ is the $n$-th partial sum of the harmonic series. (Hint: Write $\log(1-z)$ as a power series for $|z| < 1$ and use Cauchy products for $[\log(1-z)]^2$)
I've been working on this question since last night and I've been trying to follow the hint and look at the derivative of $\log(1-z)$ and then integrate that, but all attempts have been futile.
Really lost here...