Double Sequences and Double Series

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×N −→ C: 1. limn,m→∞ s(n, m), 2. limn→∞(limm→∞ s(n, m)), 3. limm→∞(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. ∑∞ n,m=1 z(n, m), 5. ∑∞ n=1( ∑∞ m=1 z(n, m)), 6. ∑∞ m=1( ∑∞ n=1 z(n, m)). ∗This research is supported by a grant from the Deanery of Scientific Research, Islamic University of Gaza