Continuous Operations and Series of Indeterminate Terms

Abstract

It is known that functions involving natural numbers are generalized to the real ones, for instance the gamma function can be viewed as a generalization of the factorial operator. In this paper, we propose to generalize the repetition of an operation over a function (composition, derivatives and integrals) toward the field of reals. It means repeating q times an operation over a function, where q is a real number. As a result, it is explained what functional and analytical dimensional extensions are and it is given a proof to theorems related to the indeterminate terms. The main finding is that every real number is expressible as a bijection of an infinite sum of elements whose coefficients are real numbers and their main values are either an indeterminate value or an infinite value. The concept of series of indeterminate values becomes relevant, as a novelty to operate with infinite, zero and indeterminate terms, which cannot be deductible from the non-standard analysis.