The problem is from Bass. I have to prove
$$\sum_{k=1}^\infty \frac{1}{(p+k)^2} = - \int_0^1 \frac{x^p}{1-x}\log x \ \mathrm{d}x$$
I am lost on this. I am trying to find some sequence I can use to use one of the convergence theorems, but I can't come up with one. Is that even an approach that would work?
$HINT$ $$\frac{x}{1-x}=\sum_{k \in \Bbb{N}} x^k ,\forall x \in [0,1)$$
Then apply Beppo-Levy Theorem to the non-negative functions on $[0,1)$ $$f_k(x)=x^{p+k-1}(-\log{x})$$
To tackle the integrals of $f_k$ make the change of variable $x=e^t$