Using known zeta-series to elucidate the origin of the Dancs-He series for $ \ln{2} $ and $ ζ{(2 n+1)}$

Details Using known zeta-series to elucidate the origin of the Dancs-He series for $ \ln{2} $ and $ ζ{(2 n+1)}$ Using known zeta-series to elucidate the origin of the Dancs-He series for $ \ln{2} $ and $ ζ{(2 n+1)}$

2009-09-28

arxiv.org/abs/0909.5234v3

Mathematics

History and Overview

7 pages, no figures. Revised and shortened. Submitted to J. T. N. Bordeaux (02/20/2010)

Scientific paper

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 applies showing that the even zeta-values are rational multiples of even powers of $ \pi$. There, 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 a more straightforward manner.

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