Closed form for series involving harmonic numbers

Question

Is there a closed form for this series values:

$$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(n)}}{k} $$

where

$$ H_k^{(n)}=\sum_{i=1}^k \frac{1}{i^n} $$

and n is a positive integer.

Thanks!

Answer 1

In this paper we have some identities for $$\alpha_{h}\left(s,t\right)=\sum_{k\geq1}\frac{\left(-1\right)^{k}}{k^{t}}H_{k-1}^{\left(s\right)} $$ (for some special value of $s $ and $t $) which can be rewritten as $$\alpha_{h}\left(s,t\right)=\left(1-2^{1-s-t}\right)\zeta\left(s+t\right)+\sum_{k\geq1}\frac{\left(-1\right)^{k}}{k^{t}}H_{k}^{\left(s\right)}. $$