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Evaluations of some variant Euler sums. (English) Zbl 1101.11033
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 $$\sum^\infty_{k=1} {h_k\over k} x^{2k}= {1\over 4}\log^2\Biggl({1+x\over 1-z}\Biggr).$$

11M41Other Dirichlet series and zeta functions
11M06$\zeta (s)$ and $L(s, \chi)$
40A05Convergence and divergence of series and sequences
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