Series with Harmonic numbers and the tail of \(\zeta(2)\)

Abstract

In this paper we solve an open problem related to the calculation of a quadratic series and we obtain that
\[
\begin{aligned}
\sum\limits_{n=1}^{\infty} H_{n}^2
\left(
\zeta(2)-1-\dfrac{1}{2^2}-\cdots-\dfrac{1}{n^2}
\right)^2
&= 6\zeta(3)-\dfrac{19}{2}\zeta(4) \\
&\quad +\dfrac{5}{2}\zeta(5)+2\zeta(2)\zeta(3).
\end{aligned}
\]

Also, we calculate the sum of the series involving the tail of \(\zeta(2)\) and the square of the \(n\)th harmonic number:
\[
\sum\limits_{n=1}^{\infty}\dfrac{H_{n}^2}{n}\left(\zeta(2)-1-\dfrac{1}{2^2}-\cdots-\dfrac{1}{n^2}\right)=2\zeta(2)\zeta(3).
\]