# Infinite Series ∑∞n=1Hnn32n\sum_{n=1}^\infty\frac{H_n}{n^32^n}

I’m trying to find a closed form for the following sum

where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number.

Could you help me with it?

In the same spirit as Robert Israel’s answer and continuing Raymond Manzoni’s answer (both of them deserve the credit because of inspiring my answer) we have

Dividing equation above by $x$ and then integrating yields

Using IBP to evaluate the green integral by setting $u=\operatorname{Li}_3(1-x)$ and $dv=\frac1x\ dx$, we obtain

Using Euler’s reflection formula for dilogarithm

then combining the blue integral in $(1)$ and $(2)$ yields

Setting $x\mapsto1-x$ and using the identity $H_{n+1}-H_n=\frac1{n+1}$, the red integral becomes

Putting all together, we have

Setting $x=1$ to obtain the constant of integration,

Thus

Finally, setting $x=\frac12$, we obtain

$[1]\$ Harmonic number
$[2]\$ Polylogarithm