Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$ Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
 A: $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}=\frac{28}{243}+\frac{10}{81} \log \left(\frac{2}{3}\right).$$
Hint: Change the order of summation:
$$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}=\sum_{n=1}^\infty\sum_{k=1}^n\frac{(-1)^n n^4}{2^n k}=\sum_{k=1}^\infty\sum_{n=k}^\infty\frac{(-1)^n n^4}{2^n k}.$$
A: A related problem. Here is another approach. Recalling the generating function of the harmonic numbers
$$ \sum_{n=1}^{\infty} H_n x^n = \frac{\ln(1-x)}{x-1} \implies (xD)^4\sum_{n=1}^{\infty} H_n x^n=(xD)^4 \frac{\ln(1-x)}{x-1}, $$
$$ \implies \sum_{n=1}^{\infty} H_n n^4 x^n = {\frac {x \left( 1+11\,x+11\,{x}^{2}+{x}^{3} \right) \ln  \left( 1-x
 \right) }{ \left( -1+x \right) ^{5}}}+{\frac {x \left( -1-27\,{x}^{2}
-18\,x-4\,{x}^{3} \right) }{ \left( -1+x \right) ^{5}}}.$$
Substituting $x=-\frac{1}{2}$ in the above identity gives the desired result
$$ \sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}={\frac {28}{243}}-{\frac {10}{81}}\,\ln  \left( \frac{3}{2} \right).  $$ 
A: Hint: Set $f_k=H_k$ and find a function $g_k$ such that  $g_{k+1}-g_k=\frac{(-1)^kk^4}{2^k}$ and use summation by parts to get:
$$\sum_{k=1}^nf_k(g_{k+1}-g_k)=f_{n+1}g_{n+1}-f_1g_1-\sum_{k=1}^n(f_{k+1}-f_k)g_{k+1}$$
$$\sum_{k=1}^nH_k(\frac{(-1)^kk^4}{2^k})=f_{n+1}g_{n+1}-f_1g_1-\sum_{k=1}^n(\frac{1}{k+1})g_{k+1}$$
Find the limit as $n\rightarrow \infty$. $g_k$ will be a function of the form $(\frac{-1}{2})^kp(k)+C$ for some polynomial $p$ and constant $C$. Thus, I believe the RHS will be a sum of the form:
$$\sum_{k=1}^{\infty}[(\frac{-1}{2})^kp(k)+C]\frac{1}{k+1}$$
which will be just tedious to evaluate.
