From this post:
The sum: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{H_n^3}{n+1}=-\frac{9}{8}\zeta(3)\ln(2)+\frac{\pi^4}{288}-\frac{\ln^4(2)}{4}+\frac{\pi^2}{8}\ln^2(2)$$
by moving the $n+1$ back to $n$
What is the sum of: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{H_n^3}{n}?$$
I guess it could be easy, but I cannot see it.
Can anyone please with this sum? I needs it to solve one of my current problem I am working on.
I will start completing Frank's answer: since $\int_{0}^{1}x^n\log(1-x)\,dx = -\frac{H_{n+1}}{n+1}$, the linear, alternating Euler sum with weight four $$ \sum_{n\geq 1}\frac{(-1)^{n-1} H_n}{n^3} $$ can be computed from the integral $\int_{0}^{1}\text{Li}_2(-x)\log(1-x)\frac{dx}{x}$ or from the integral (equivalent by $\text{IBP}$) $\int_{0}^{1}\text{Li}_2(x)\log(1+x)\frac{dx}{x}$. Up to a $\eta(4)=\frac{7\pi^4}{720}$ term, this is the same as computing $$ \int_{0}^{1}\log^2(1+x)\log^2(x)\frac{dx}{x}\quad\text{or}\quad\int_{0}^{1}\log(1+x)\log^3(x)\frac{dx}{1+x}$$ since $\log^2(1-x)=\sum_{n\geq 1}\frac{2H_{n-1}}{n}x^n$. In terms of the notation of Flajolet and Salvy, we are tackling $S_{1,3}^{+-}$, which is given by $$ \mu_1=\frac{1}{2}\int_{0}^{1}\frac{\log^2(z)\log(1+z)}{z(1+z)}\,dz $$ or $$ \mu_1=\frac{11 \pi ^4}{360}+\frac{\pi^2}{12}\log^2(2)-\frac{1}{12}\log(2)^4-2\,\text{Li}_4\left(\frac{1}{2}\right)-\frac{7}{4}\log(2)\zeta(3)$$ (De Doelder 1991). It remains to crack the quadratic, alternating Euler sum with weight four $$ \sum_{n\geq 1}\frac{(-1)^{n+1}H_n^2}{n^2} $$ where $$ \sum_{n\geq 0}(-1)^n\frac{H_n^{(2)}}{(n+1)^2} = \int_{0}^{1}\frac{\text{Li}_2(-x)}{1+x}(-\log(x))\,dx $$ is related to $\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^2}\sum_{m\geq 1}\frac{1}{m^2}$ and $\pi^4$ via symmetry and summation by parts, while $$ \sum_{n\geq 1}\frac{H_{n-1}^2-H_{n-1}^{(2)}}{n}x^n = -3\log^3(1-x)$$ by the structure of the Stirling numbers of the first kind. Therefore the computation can be finished by noticing that $$ \int_{0}^{1}\log^3(1-x)\frac{dx}{x} = -\frac{\pi^4}{15}. $$ Mathematica's command $\text{FindIntegerNullVector}$ returns $$ \sum_{n\geq 1}\frac{(-1)^{n+1}H_n^3}{n}=\\=\frac{1}{90} (68 -\pi^4+ 12\pi^2\log^2(2)- 197\log(2)^4+ 47\,\text{Li}_4(1/2)+ 22\log(2)\zeta(3))$$ as a plausible identity.
Here might be a start. Observe that$$\begin{align*}\sum\limits_{n\geq1}(-1)^{n+1}\frac {H_n^3}{n+1} & =\frac {H_1^3}2-\frac {H_2^3}3+\frac {H_3^3}4-\cdots\\ & =\sum\limits_{n\geq2}(-1)^n\frac {H_{n-1}^3}n\end{align*}$$
Now use the fact that $H_n=H_{n-1}+\tfrac 1n$. Therefore, the original sum becomes$$\begin{align*}\sum\limits_{n\geq1}(-1)^{n+1}\frac {H_n^3}{n+1} & =\sum\limits_{n\geq2}(-1)^n\frac {1}n\left[H_n-\frac 1n\right]^3\\ & =\sum\limits_{n\geq2}(-1)^n\frac 1n\left[H_n^3-\frac {3H_n^2}n+\frac {3H_n}{n^2}-\frac 1{n^3}\right]\\ & =\sum\limits_{n\geq2}(-1)^{n}\frac {H_n^3}{n}-3\sum\limits_{n\geq2}(-1)^n\frac {H_n^2}{n^2}+3\sum\limits_{n\geq2}(-1)^{n}\frac {H_n}{n^3}+\frac {7\pi^4}{720}\\ & =-\sum\limits_{n\geq2}(-1)^{n+1}\frac {H_n^3}n+3\sum\limits_{n\geq2}(-1)^{n+1}\frac {H_n^2}{n^2}+3\sum\limits_{n\geq2}(-1)^{n}\frac {H_n}{n^3}+\frac {7\pi^4}{720}\end{align*}$$
The second sum can be found with the help of this question. Reindex the identity to start the sum from two to get
$$\sum\limits_{n\geq2}(-1)^n\frac {H_n}{n^3}\color{red}{=1+2\operatorname{Li}_4\left(\frac 12\right)+\frac {7\log 2}4\zeta(3)-\frac {11\pi^4}{360}+\frac {\log^42}{12}-\frac {\pi^2\log^22}{12}}$$
In a similar fashion, some digging on MSE gives this question which might be a big help in tackling the second sum:$$\sum\limits_{n\geq2}(-1)^n\frac {H_n^2}{n^2}$$
