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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 \[ \sum^\infty_{k=1} {h_k\over k} x^{2k}= {1\over 4}\log^2\Biggl({1+x\over 1-z}\Biggr). \]

MSC:

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
40A05 Convergence and divergence of series and sequences

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