In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski  in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.
Applied Mathematics | Mathematics
He, Tian-Xiao and Dancs, Michael, "Numerical approximation to ζ(2n+1)" (2006). Scholarship. 6.