Using known zeta-series to derive the Dancs-He series for $\,\ln{2}\,$ and $\,\zeta{(2\,n+1)}$

TL;DR

This paper presents an alternative derivation of the Dancs-He series for  and  using known zeta-series, providing new insights into their structure and relation to known mathematical series.

Contribution

It introduces a novel method to derive the Dancs-He series for  and 3 using known zeta-series, differing from previous manipulation-based approaches.

Findings

Derived the Dancs-He series using known zeta-series.

Provided an alternative perspective on the appearance of  and 3.

Enhanced understanding of series representations for  and 3.

Abstract

In a recent work, Dancs and He found new `Euler-type' formulas for $\,\ln{2}\,$ and $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, each containing a series that apparently can not be evaluated in closed form, distinctly from $\,\zeta{(2\,n)}$, for which the Euler's formula allows us to write it as a rational multiple of $\,\pi^{2n}$. There in that work, however, the formulas are derived through certain series manipulations, by following Tsumura's strategy, which makes it \emph{curious} --- in the words of those authors themselves --- the appearance of the numbers $\,\ln{2}\,$ and $\,\zeta{(2\,n+1)}$. In this short paper, I show how some known zeta-series can be used to derive the Dancs-He series in an alternative manner.