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

Abstract

In a recent work, Dancs and He found new `Euler-type' formulas for $\ln 2$ and $\zeta (2n+1)$, $n$ being a positive integer, each containing a series that apparently can not be evaluated in closed form, distinctly from $\zeta (2n)$, 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 $\zeta (2n+1)$. In this short paper, I show how some known zeta-series can be used to derive the Dancs-He series in an alternative manner.

arXiv comment7 pages, no figures. Second revision completed. Only small corrections. Submitted to J. T. N. Bordeaux (04/07/2011)

References in corpus (2)

• Notes on a paper of Tyagi and Holm: A new integral representation for the Riemann Zeta function
• A new integral representation for the Riemann Zeta function