Define the sum $$S_k = \sum_{n=1}^\infty\frac{1}{n(n+k)}$$ for all integers $k\ge0$. We know that $S_0=\dfrac{\pi^2}{6}$ and $S_1=1$. Is it possible to write out $S_k$ explicitly for larger values of $k$, or at the very least give some asymptotic expression for $k$? All I actually need to know is whether $S=(S_k)_{k=0}^\infty\in\ell^2$, but I'd be interested in the answers to the previous questions as well.
(My motivation for this is to show that the operator that satisfies $\langle A e_i,e_j\rangle_{\ell^2(\mathbb N)} = \frac{1}{i+j-1}$ is not bounded on $\ell^2$, showing that Schur's condition does not hold if we replace $\ell^1$-boundedness with $\ell^2$-boundedness)
For $0\not=k\in\mathbb{N}$ the series telescopes:
$$S_k={1\over k}\sum_{n=1}^\infty\left({1\over n}-{1\over n+k}\right)={1\over k}\left(1+{1\over2}+\cdots+{1\over k}\right)\approx{\log k\over k}$$
so the sum $\sum S_k^2$ converges.