<s>[INST] Series $\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2^8}-\frac{1}{3^8}\cdots\frac{1}{n^8}\right]$ How can we find the sum of  this hard series any hint please 
My trial is to express zeta(8) as a series but I don't know how and what to do
$$\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2^8}-\frac{1}{3^8}\cdots\frac{1}{n^8}\right]$$
 [/INST] We shall begin by calculating the multivariate zeta value $\zeta(7,1)$ so we have it to hand later. Using "well known formula" and Euler reflection formula, we have
\begin{eqnarray*}
\zeta(8)&=&\zeta(7,1)+\zeta(6,2)+\zeta(5,3)+\zeta(4,4)+\zeta(3,5)+\zeta(2,6) \\
\zeta(2)\zeta(6)&=&\zeta(8)+\zeta(6,2)+\zeta(2,6) \\
\zeta(3)\zeta(5)&=&\zeta(8)+\zeta(5,3)+\zeta(3,5) \\
\zeta(4)^2&=&\zeta(8)+2\zeta(4,4). \\
\end{eqnarray*}
Linear algebra gives $\zeta(7,1)=\frac{1}{2}(7\zeta_8-2\zeta_6\zeta_2 -2\zeta_5\zeta_3 -\zeta_4^2)$ & recall that $\zeta_2=\frac{\pi^2}{6},\zeta_4=\frac{\pi^4}{90},\zeta_6=\frac{\pi^6}{945},\zeta_8=\frac{\pi^8}{9450}$.
OK ... the question 
\begin{eqnarray*}
\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2^8}-\frac{1}{3^8}\cdots\frac{1}{n^8}\right] 
= \sum_{n=1}^{\infty} \sum_{m=n+1}^{\infty} \frac{H_n}{m^8} \\
= \sum_{n=1}^{\infty} \sum_{i=1}^{\infty} \frac{H_n}{(i+n)^8}  = \sum_{n=1}^{\infty} \sum_{i=1}^{\infty} \sum_{j=1}^{n}\frac{1}{j(i+n)^8} 
\end{eqnarray*}
Invert the order of the $j$ and $n$ plums, substitute $n=j+k-1$, 
\begin{eqnarray*}
\sum_{i=1}^{\infty} \sum_{n=1}^{\infty} \sum_{j=1}^{n}\frac{1}{j(i+n)^8} =\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{n=j}^{\infty}\frac{1}{j(i+n)^8} =\sum_{i=1}^{\infty} \sum_{j=1}^{\infty}  \sum_{k=1}^{\infty}\frac{1}{j(i+j+k-1)^8} 
\end{eqnarray*}
substitute $p=i+k-1$ & partial fractions $\frac{p}{j(j+p)}=\frac{1}{j}-\frac{1}{p+j}$
\begin{eqnarray*}
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{k=1}^{\infty}\frac{1}{j(i+j+k-1)^8} =\sum_{p=1}^{\infty} \sum_{j=1}^{\infty}\frac{p}{j(p+j)^8} =\sum_{p=1}^{\infty}  \sum_{j=1}^{n}\frac{1}{j(p+j)^7}-\sum_{p=1}^{\infty}  \sum_{j=1}^{n}\frac{1}{(p+j)^8} 
\end{eqnarray*}
note that the penultimate sum is $\zeta(7,1)$ & substitute $q=p+j$ in the last sum
\begin{eqnarray*}
\sum_{p=1}^{\infty}  \sum_{j=1}^{n}\frac{1}{(p+j)^8} = \sum_{q=1}^{\infty}  \frac{q-1}{q^8} = \zeta(7)- \zeta(8).
\end{eqnarray*}
And so we have 
\begin{eqnarray*}
\sum_{n=1}^{\infty} H_n\left[\zeta(8)-\frac{1}{1^8}-\frac{1}{2^8}-\frac{1}{3^8}\cdots\frac{1}{n^8}\right] = \color{red}{\frac{\pi^8}{4200}-\zeta(3)\zeta(5)-\zeta(7)}.
\end{eqnarray*}
</s>