# A sum of six alternating harmonic series

In one of the answers to this question Evaluate $\int_0^1\arcsin^2(\frac{\sqrt{-x}}{2}) (\log^3 x) (\frac{8}{1+x}+\frac{1}{x}) \, dx$, the OP also presents some results from a paper, many of them obviously related to sums of harmonic series of weight $$6$$ (perhaps also magical cancellations involving some of the closed forms prevent me from saying more).

Here is an example of a sum that is strongly related to those sums - in fact, to solve those questions elementarily one wants to get transformations (this may take some time and it can be a boring task) to such forms as the one below (this is just one example)

$$\begin{equation*} \sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n}{n^5} -\frac{3}{2}\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^2}{n^4} +\frac{4}{3}\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^3}{n^3}-\frac{2}{3}\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^4}{n^2} \end{equation*}$$ $$\begin{equation*} +\frac{1}{6}\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^{(3)}}{n^3}-\frac{1}{3}\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n H_n^{(3)}}{n^2} \end{equation*}$$ $$\begin{equation*} =\frac{19}{128} \zeta (6)+\frac{1}{16}\zeta^2(3). \end{equation*}$$

Since the previous question with such questions had been well received, maybe the community will enjoy some more sums of series alike, leading to very simple and neat closed forms.

The sum of the series above is proposed by C. I. Valean, and it is obtained very elegantly - such a solution will be presented here after the release of More (Almost) Impossible Integrals, Sums, and Series, the sequel of (Almost) Impossible Integrals, Sums, and Series.

How would you attack it?

• "the OP also presents some results from a paper...." which paper precisely? Commented Apr 2, 2023 at 11:48
• @pisco You referred to arxiv.org/abs/1010.4298 in your answer at the given MSE link, right in the beginning, "This paper of 2010 conjectured many infinite sums. The following are the most difficult among them:" Commented Apr 3, 2023 at 11:13
• I see. Except for a few of them, many identities conjectured there (all proved now) are not quite on the same level of difficulty as those you posted on this site or in your books. arxiv.org/pdf/2210.07238.pdf contains some new conjectures (some are also proved now) of the same author, you might be interested in them. Commented Apr 3, 2023 at 11:25
• The identity of this question seems quite unrelated to those papers. All 6 sums can be expressed in terms of $\text{Li}_n(1/2)$ or $\text{Li}_n((1+i)/2)$. Although in my proof of math.stackexchange.com/questions/4592985, one needs to evaluate thousand to million such combinations you asked. Commented Apr 3, 2023 at 11:38
• @pisco As regards the series presented in the paper, I'll share more after finalizing some of the work on them. As long as I enjoy a problem, no level of (suggested) difficulty can ever prevent me from giving it a try (and I never give up easily). Commented Apr 3, 2023 at 12:14