On General Sum Approximations of Irrational Numbers.

There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these representations yield the same class of real numbers. If we work with some restricted notion of computability, e.g., polynomial time computability or primitive recursive computability, they do not. This phenomenon has been investigated over the last seven decades by Specker [13], Mostowski [8], Lehman [10], Ko [3, 4], Labhalla and Lombardi [9], Georgiev [1], Kristiansen [5, 6] and quite a few more. Georgiev et al. [2] is an extended version of the current paper.Irrational numbers can be represented by infinite sums. Sum approximations from below and above were introduced in Kristiansen [ 5] and studied further in Kristiansen [ 6]. Every irrational number (alpha )