Double sequences and double series

Abstract EN

This research considers two traditional important questions, which are interesting, at least to most mathematicians.

The first question arises in the theory of double sequences of complex numbers, which concerns the relationship, if any, between the following three limits of a double sequence s : N x N —► C: 1.

limn, m→∞s(n, m), 2.

limn →∞ (lim m→∞ s (n, m)), 3.

lim m→∞ (limn→∞S(n, m)).

In particular, we’ll address the question of when can we interchange the order of the limit for a double sequence {s (n, m)}; that is, when the limit (2) above equals the limit (3) above.

The answer to this question is found in Theorem 2.13.

The second question arises in the theory of double series of complex numbers, which concerns the relationship, if any, between the following series : 4.

Σn∞, m=1 z(n, m), 5.

Σn∞=1 (Σm∞=1 z(n, m)), 6.

Σm∞=1 (Σn∞=1 z(n, m)).

In particular, we’ll address the question of when can we interchange the or¬der of summation in a doubly indexed infinite series; that is, when the series (5) above equals the series (6) above.

The answers to this question are found in Theorems 7.5, 8.6, and 9.5.

The topics of the above-mentioned two questions have not received enough attention within the mathematical community, so that there has been scattered answers in the literature (see [1-7]).

Up to this moment, one can’t find a single textbook or a research paper that gives a full account to such topics.

In this technical article, we’ll, among other things, attempt to give such an expository account which will summarize facts from the basic theory of double sequences and double series and gives detailed proofs of them.

Many of the results collected are well known and can be found in the supplied references.

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