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!