<s>[INST] Evaluate $1+\left(\frac{1+\frac12}{2}\right)^2+\left(\frac{1+\frac12+\frac13}{3}\right)^2+\left(\frac{1+\frac12+\frac13+\frac14}{4}\right)^2+...$ Evaluate:
$$S_n=1+\left(\frac{1+\frac12}{2}\right)^2+\left(\frac{1+\frac12+\frac13}{3}\right)^2+\left(\frac{1+\frac12+\frac13+\frac14}{4}\right)^2+...$$
a_n are the individual terms to be summed.
My Try :
\begin{align}
&a_1=1\\
&a_2=\left(\frac{3}{4}\right)^2=\frac{9}{16}\\
&a_3=\left(\frac{11}{18}\right)^2\\
&a_4=\left(\frac{25}{48}\right)^2
\end{align}
now :?
 [/INST] Recall that the multiple zeta values are defined by the series
$$
\zeta(s_1,\ldots,s_k):=\sum_{n_1>\ldots>n_k\geq 1}\frac{1}{n_1^{s_1}\ldots n_k^{s_k}}.
$$
The sum $S$ can be expressed as a linear combination of multiple zeta values. We have
$$
\begin{align*}
S&=\sum_{n=1}^\infty \sum_{k_1,k_2=1}^n\frac{1}{n^2k_1k_2}\\
&=\left(2\sum_{n>k_1>k_2}+\sum_{n>k_1=k_2}+2\sum_{n=k_1>k_2}+\sum_{n=k_1=k_2}\right)\frac{1}{n^2k_1k_2}\\
&=2\zeta(2,1,1)+\zeta(2,2)+2\zeta(3,1)+\zeta(4).
\end{align*}
$$
Each of these multiple zeta values is a rational multiple of $\pi^4$. The expressions have been tabulated for instance on the MZV data mine:
$$
\begin{align*}
\zeta(2,1,1)&=\frac{\pi^4}{90},\\
\zeta(2,2)&=\frac{\pi^4}{120},\\
\zeta(3,1)&=\frac{\pi^4}{360},\\
\zeta(4)&=\frac{\pi^4}{90}.
\end{align*}
$$
So we get
$$
S=\frac{17\pi^4}{360}
$$
</s>