How to evaluate $$\sum_{n=1}^{\infty}\frac{H_n}{n^52^n}~~,~~\sum_{n=1}^{\infty}\frac{H_n}{n^62^n}$$ In this post M gives a closed form for the first series,but it contains a zeta function,can it be simplified any more? And is there a closed form for the second series.
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