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: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$

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.
