# Explicit evaluation of Euler sums

Borwein, David and Borwein, Jonathan M. and Girgensohn, Roland (1995) Explicit evaluation of Euler sums. Proceedings of the Edinburgh Mathematical Society (38). pp. 277-294.

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Postscript
In response to a letter from Goldbach, Euler considered sums of the form $$\sh(s,t):=\sum\limits_{n=1}^\infty (1+{1 \over 2^s}+\dots+{1 \over (n-1)^s}) \, n^{-t},$$ where $s$ and $t$ are positive integers. As Euler discovered by a process of extrapolation (from $s+t \le 13$), $\sh(s,t)$ can be evaluated in terms of Riemann $\zeta$-functions when $s+t$ is odd. We provide rigorous proof of Euler's discovery and then give analogous evaluations with proof for corresponding alternating sums. Relatedly we give a formula for the harmonic sum $$\sum\limits_{n=1}^\infty (1+{1 \over 2}+\dots+{1 \over n})^2 \, (n+1)^{-m}.$$ This evaluation involves $\zeta$-functions and $\sh(2,m).$
Item Type: Article pubdom FALSE Riemann zeta function, beta function, psi function, generating functions, polylogarithms, harmonic numbers 40-xx Sequences, series, summability > 40Axx Convergence and divergence of infinite limiting processes11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory40-xx Sequences, series, summability > 40Bxx Multiple sequences and series33-xx Special functions > 33Exx Other special functions UNSPECIFIED Users 1 not found. 16 Nov 2003 07 Sep 2014 20:26 https://docserver.carma.newcastle.edu.au/id/eprint/58