asymptotic estimate of $\sum 1/(k^2 H_k)$ where $H_k$ is the harmonic series

Question

I need some help here. I started with $$\sum_{k=1}^N \frac{1}{k^2 H_k}=\sum_{k=1}^N \frac{1}{k^2 \int_0^1\frac{1-x^k}{1-x}dx}=\sum_{k=1}^N \frac{1}{\int_0^1 k^2 \frac{1-x^k}{1-x}dx}$$

But I do not know how to continue anymore.

Answer 1

Hint. Note that Harmonic numbers have a nice asymptotic estimate $$H_n\approx\ln n+\gamma+o(1)$$ where $\gamma\approx 0.5772156649$ is the Euler-Mascheroni constant. Moreover for $n\geq 1$, $$\frac{1}{2}<H_n-\ln n\leq 1.$$