Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$
Can anyone help me please?

Solutions Collecting From Web of "Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$"

Let’s start with the product of $\;-\ln(1-x)\,$ and $\dfrac 1{1-x}$ to get the product generating function
(for $|x|<1$) :
$$\tag{1}f(x):=-\frac {\ln(1-x)}{1-x}=\sum_{n=1}^\infty H_n\, x^n$$
Dividing by $x$ and integrating we get :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}n\, x^n&=\int \frac{f(x)}xdx\\
&=-\int \frac{\ln(1-x)}{1-x}dx-\int\frac{\ln(1-x)}xdx\\
\tag{2}&=C+\frac 12\ln(1-x)^2+\operatorname{Li}_2(x)\\
\end{align}
(with $C=0$ from $x=0$)
The first integral was obtained by integration by parts, the second from the integral definition of the dilogarithm or the recurrence for the polylogarihm (with $\;\operatorname{Li}_1(x)=-\ln(1-x)$) : $$\tag{3}\operatorname{Li}_{s+1}(x)=\int\frac {\operatorname{Li}_{s}(x)}x dx$$

Dividing $(2)$ by $x$ and integrating again returns (using $(3)$ again) :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n&=\int \frac {\ln(1-x)^2}{2\,x}dx+\int \frac{\operatorname{Li}_2(x)}x dx\\
&=C+I(x)+\operatorname{Li}_3(x)\\
\end{align}
with $I(x)$ obtained by integration by parts (since $\frac d{dx}\operatorname{Li}_2(1-x)=\dfrac {\ln(x)}{1-x}$) :
\begin{align}
I(x)&:=\int \frac {\ln(1-x)^2}{2\,x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}\right|+\int \ln(1-x)\frac {\ln(x)}{1-x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)\right|+\int \frac{\operatorname{Li}_2(1-x)}{1-x}dx\\
&=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)-\operatorname{Li}_3(1-x)\right|\\
\end{align}
getting the general relation :
$$\tag{4}\sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n=C+\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)+\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)$$
(with $C=\operatorname{Li}_3(1)=\zeta(3)$ here)
applied to $x=\dfrac 12$ with $\operatorname{Li}_2\left(\frac 12\right)=\dfrac{\zeta(2)-\ln(2)^2}2$ from the link returns the wished :
\begin{align}
\sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\frac{\ln(2)^3}2-\ln(2)\frac{\zeta(2)-\ln(2)^2}2\\
\tag{5}\sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\ln(2)\frac{\zeta(2)}2
\end{align}

We will make frequent use of
$$
\binom{n+1}{k+1}=\binom{n}{k}\frac{n+1}{k+1}\tag{1}
$$
The Generalized Harmonic Numbers of the second order are defined as
$$
H_n^{(2)}=\sum_{k=1}^n\frac1{k^2}\tag{2}
$$
The factor of $2^{-n}$ in each term reminded me of the Euler Series Transformation. Reversing the series acceleration (series deceleration?), it can be seen that we should look at
$$
\begin{align}
\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1}
&=\sum_{k=0}^n\binom{n}{k}\frac{(-1)^k}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{3}\\
&=\frac1{n+1}\sum_{k=0}^n(-1)^k\binom{n+1}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{4}\\
&=\frac1{n+1}\sum_{j=0}^n\sum_{k=j}^n(-1)^k\binom{n+1}{k+1}\frac1{(j+1)^2}\tag{5}\\
&=\frac1{n+1}\sum_{j=0}^n(-1)^j\binom{n}{j}\frac1{(j+1)^2}\tag{6}\\
&=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\binom{n+1}{j+1}\frac1{j+1}\tag{7}\\
&=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\sum_{k=0}^n\binom{k}{j}\frac1{j+1}\tag{8}\\
&=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\sum_{j=0}^n(-1)^j\binom{k+1}{j+1}\tag{9}\\
&=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\tag{10}\\[3pt]
&=\frac{H_{n+1}}{(n+1)^2}\tag{11}
\end{align}
$$
Explanation:
$\ \;(3)$: use $(2)$
$\ \;(4)$: apply $(1)$
$\ \;(5)$: change order of summation
$\ \;(6)$: $\sum\limits_{j=k}^n(-1)^j\binom{n+1}{j+1}=(-1)^k\binom{n}{k}$
$\ \;(7)$: apply $(1)$
$\ \;(8)$: $\sum\limits_{j=0}^n\binom{j}{k}=\binom{n+1}{k+1}$
$\ \;(9)$: $\sum\limits_{j=0}^k(-1)^j\binom{k+1}{j+1}=1$
$(10)$: $H_{n+1}=\sum\limits_{k=0}^n\frac1{k+1}$

Using $(11)$, the Euler Series Transformation says that
$$
\begin{align}
\sum_{k=0}^\infty(-1)^k\frac{H_{k+1}^{(2)}}{k+1}
&=\sum_{n=0}^\infty2^{-n-1}\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1}\\
&=\sum_{n=0}^\infty2^{-n-1}\frac{H_{n+1}}{(n+1)^2}\tag{12}
\end{align}
$$
Therefore, with a change of indexing, we get
$$
\begin{align}
\sum_{n=1}^\infty2^{-n}\frac{H_n}{n^2}
&=\sum_{n=1}^\infty(-1)^{n-1}\frac{H_n^{(2)}}{n}\tag{13}\\
&=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3}+\sum_{n=1}^\infty(-1)^{n-1}\frac{H_{n-1}^{(2)}}{n}\tag{14}\\
&=\frac34\zeta(3)+\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sum_{k=1}^{n-1}\frac1{k^2}\tag{15}\\
&=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^{n-1}}{nk^2}\tag{16}\\
&=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{(-1)^{k+n-1}}{(k+n)k^2}\tag{17}\\
&=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty(-1)^{k+n-1}\left(\frac1{k^2n}-\frac1{kn(k+n)}\right)\tag{18}\\[6pt]
&=\frac34\zeta(3)-\frac12\zeta(2)\log(2)+\frac14\zeta(3)\tag{19}\\[9pt]
&=\zeta(3)-\frac12\zeta(2)\log(2)\tag{20}
\end{align}
$$
Explanation:
$(13)$: reindex $(12)$
$(14)$: $H_n^{(2)}=\frac1{n^3}+H_{n-1}^{(2)}$
$(15)$: apply $(2)$
$(16)$: change order of summation
$(17)$: reindex $n\mapsto k+n$
$(18)$: $\frac1{(k+n)k^2}=\frac1{k^2n}-\frac1{kn(k+n)}$
$(19)$: $\sum\limits_{k=1}^\infty\sum\limits_{n=1}^\infty\frac{(-1)^{k+n}}{kn(k+n)}=\frac14\zeta(3)$ from $(5)$ and $(7)$ of this answer
$(20)$: addition

\begin{eqnarray}
\sum\limits_{n=1}^\infty \frac{H_n}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \sum\limits_{n=m}^\infty \frac{1}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \int\limits_{-\infty}^0(-\xi) \frac{(1/2 \exp(\xi))^m}{1-1/2 \exp(\xi)} d\xi = \\
\int\limits_{-\infty}^0 \xi \frac{\log(1 – 1/2 \exp(\xi))}{1-1/2 \exp(\xi)} d\xi = \\
\int\limits_{1/2}^1 \left(\frac{1}{u} + \frac{1}{1-u}\right) \log(u) \left[\log(2) + \log(1-u)\right] du = \\
\zeta(3) – \frac{1}{12} \pi^2 \log(2)
\end{eqnarray}

I think that all the steps are clear except for the last two ones.In the second last step I substituted for 1 – 1/2 exp(xi).The only non-trivial integrals in here are $\int \log(u)/(1-u) du$ and $\int \log(u) \log(1-u)/(1-u) du$. I compute them now.
The first integral is done by expanding the denominator in a series and integrating term by term.
\begin{equation}
\int\limits_{1/2}^1 \frac{\log(u)}{1-u} du = \sum\limits_{p=0}^\infty \int\limits_{1/2}^1 u^p \log u du = \sum\limits_{p=0}^\infty \frac{-1+2^{-1-p}}{(p+1)^2} + \log(2) \sum\limits_{p=0}^\infty \frac{2^{-1-p}}{p+1} = -\zeta(2) + Li_2(1/2) + \log(2) Li_1(1/2)
\end{equation}
The second integral is done by integrating by parts and using the definition of the polylogarithmic function.
\begin{equation}
\int\limits_{1/2}^1 \log(u) \frac{\log(1-u)}{1-u} du = \int\limits_{1/2}^1 \log(1-u) Li_2^{‘}(1-u) du = \left.\log(1-u) Li_2(1-u)\right|_{1/2}^1 + Li_3(1/2) = \log(2) Li_2(1/2) + Li_3(1/2)
\end{equation}
Now, the only thing that remains is to bring the results together. I am sorry but due to time constraints I am not able to do it right now. I have verified with Mathematica that all the partial results are correct.

Final Note: We can clearly see that the result is expressed though elementary functions and through polylogarithms of order not bigger than three, evaluated at 1/2.From the Wikipedia page on Polylogarithms we learn those polylogarithms at 1/2 are expressed in closed form through $\pi$, $\log(2)$ and the $\zeta$ functions. Having said that we can say that this completes the proof.