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Zbl 1101.11033
Chen, Hongwei
Evaluations of some variant Euler sums.
(English)
[J] J. Integer Seq. 9, No. 2, Article 06.2.3, 9 p., electronic only (2006). ISSN 1530-7638/e

This paper obtains closed form evaluations of various sums of the form $\sum^\infty_{k=1} a_k h_k$, where $$h_k= \sum^k_{r=1}(2^r- 1)^{-1}$$ and $a_k$ is a simple function of $k$. Many of them are consequences of the power series expansion $$\sum^\infty_{k=1} {h_k\over k} x^{2k}= {1\over 4}\log^2\Biggl({1+x\over 1-z}\Biggr).$$
MSC 2000:
*11M41 Other Dirichlet series and zeta functions
11M06 Riemannian zeta-function and Dirichlet L-function
40A05 Convergence of series and sequences

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