Here http://integralsandseries.prophpbb.com/topic119.html
We came across the following harmonic sum
$$\tag{1} \sum_{k\geq 1}\frac{(-1)^{k-1}}{k^2}H_k^{(2)}$$
Note that we define
$$H_k^{(2)}=\sum_{n\geq 1}^k\frac{1}{n^2} $$
Also we have
$$\psi_1(k+1)= \zeta(2) -H_k^{(2)} $$
Any ideas how to evaluate (1) ?