Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for that generalizes Apéry's series for , and appears to give the best possible series relations of this type. The formula reduces to a finite but apparently nontrivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a formula that provides strange evaluations of a large new class of nonterminating hypergeometric series.