Explicit evaluation of Euler sums

In response to a letter from Goldbach, Euler considered sums of the form [unable to replicate formula] where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t ≥ 13), σ_h(s,t) can be evaluated in terms of Riemann ζ-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 [unable to replicate formula]. This evaluation involves ζ-functions and σ_h(2,m)