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Explicit evaluation of Euler sums. (English) Zbl 0819.40003
In response to a letter from Goldbach, Euler considered sums of the form \[ \sigma_ h (s,t):= \sum_{n=1}^ \infty \Biggl(1+ {1\over {2^ s}}+ \dots+ {1\over {(n-1)^ s}} \Biggr) n^{-t}, \] where \(s\) and \(t\) are positive integers.
As Euler discovered by a process of extrapolation (from \(s+t\leq 13\)), \(\sigma_ h (s,t)\) can be evaluated in terms of Riemann \(\zeta\)- functions when \(s+t\) is odd. We provide a rigorous proof of Euler’s discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we give a formula for the series \[ \sum_{n=1}^ \infty \bigl(1+ {\textstyle {{1\over 2}+\dots+ {1\over n}}}\bigr)^ 2 (n+1)^{-m}. \] This evaluation involves \(\zeta\)- functions and \(\sigma_ h (2,m)\).

MSC:
40A25 Approximation to limiting values (summation of series, etc.)
40B05 Multiple sequences and series
11M99 Zeta and \(L\)-functions: analytic theory
33E99 Other special functions
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