A series involving harmonic numbers

Question 1

Does anyone know the exact value of this:
$\sum_{k=1}^{\infty} (-1)^k\frac{H_k}{k}$
or this:
$\sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(2)}}{k}$
Thanks!

Thanks again for the answers! I found very interesting that the integral gives exact values up to r=3 but from 4 this integral gives not exact values:

$\int_{-1}^0 \frac{Li_4(t)}{t(1-t)} \mathrm{d}t$

because wolfram says that “no results found in terms of standard mathematical functions”

Question 2

From my answer in my previous avatar, denoting
$A(p,q) = \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}H_k^{(p)}}{k^q}$
we have
$A(1,1) = \dfrac{\zeta(2)-\log^2(2)}2$
and
$A(2,1) = \zeta(3) - \dfrac{\zeta(2)\log(2)}2$

A series involving harmonic numbers

Answer

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