Closed form for series involving harmonic numbers

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!