### Generalization of the notions of a numerical series and infinite product

#### Abstract

The purpose of this article is generalizing of a series concept, an infi nite product concept, and their convergence by defining a new general infinite operation concept which is defined for any metric space and for any binary operation.

The most important special cases of an infinite operation are a series and an infinite product, however, infi nite unions, intersections and function compositions are also considered in the article.

The main result of the article is a proof of the general necessary condition of infinite operation convergence which asserts that the terms of a convergent infinite operation limit to the neutral

element under certain assumptions.

The most important special cases of an infinite operation are a series and an infinite product, however, infi nite unions, intersections and function compositions are also considered in the article.

The main result of the article is a proof of the general necessary condition of infinite operation convergence which asserts that the terms of a convergent infinite operation limit to the neutral

element under certain assumptions.

#### Keywords

Binary operation; Metric space; Convergence

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PDF (Українська)#### References

Dorogovcev, A. Y. (1989). Elements of the general theory of measure and integral [in Russian]. Kyiv: Vyshcha shkola.

Ilin, V. A., & Poznyak, Y. G. (2005). Fundamentals of mathematical analysis [in Russian]. Moscow: Fizmatlit.

Kolmogorov, A. N., & Fomin, S. V. (1957). Elements of the theory of functions and functional analysis. Rochester, NY: Graylock Press.

Kurosh, A. G. (1972). Higher algebra. Moscow: Mir Publishers.

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