What is the Definition of an Infinite Double Series? And Does Such a Series Always Have a Finite Sum?

The question of whether an infinite double series, specifically marked as an infinite double series, always has a finite sum is intriguing and deeply rooted in the theory of infinite series. This article aims to explore this concept along with the nuances of rearranging the terms of such series, especially when dealing with the structure of a double sum over the natural numbers.

Definition of an Infinite Double Series

Consider the expression of an infinite double series, [sum_{Asubseteqmathbb{N}^2} f_{mn},]

where (f_{mn}) are the terms of the series and (A) is an arbitrary subset of the natural numbers squared, (mathbb{N}^2). The straightforward method to approach such a series is to sum the rows first and then the columns, or vice versa. However, this approach alone may not always provide a clear or consistent sum, leading to the importance of the order and method of summation.

The Role of Order in Summation

The concept of order plays a critical role in the summation of infinite double series. In single-dimensional infinite series (i.e., series over the natural numbers, (mathbb{N})), the natural order of adding the entries of a sequence (a_m) is clear. Ambiguity primarily arises in conditionally convergent sums. However, in the context of a double sum over (mathbb{N}^2), the initial order of adding the terms (f_{mn}) is not immediately evident. Different methods of summation can lead to different results, reflecting the complexity of such series.

Rearranging Terms and Absolute Convergence

The issue of rearranging terms in an infinite double series is more complex. Different methods of rearranging the terms can yield entirely different sums, which can be either finite or infinite depending on the nature of the series. For instance, if we sum the rows first, we may get a finite sum, whereas summing the columns first might result in an infinite sum. This highlights the importance of the convergence properties of the series, particularly absolute convergence.

Convergence and Absolute Convergence

Convergence in infinite series is a fundamental concept. A series is said to converge if the sequence of partial sums approaches a finite limit as the number of terms increases. This limit is the sum of the series. However, absolute convergence is a stronger condition, where the series of the absolute values of the terms also converges. It is a well-known result in analysis that if a series converges absolutely, then the sum is not affected by the order of the terms. This is a crucial property for infinite series and ensures that the value of the sum is well-defined and independent of the order of summation.

Rigging a Series to Demonstrate Rearrangement

To demonstrate the impact of rearrangement, consider a specific function (f_{mn}) where summing the rows first yields a finite sum, but summing the columns first results in an infinite sum. This can be achieved by a sophisticated construction where the terms are strategically placed to exploit the order of summation. The example illustrates that without absolute convergence, the sum of an infinite series can be sensitive to the order of summation, which makes the value of the series unpredictable and unreliable.

The Importance of Order

These examples underscore the importance of understanding and insisting on the concept of absolute convergence. When dealing with infinite series, particularly those involving multiple dimensions, the order of summation can significantly affect the result. Absolute convergence guarantees that the sum is well-defined and robust against changes in the order of summation.

In conclusion, the intricacies of infinite double series and how they behave under rearrangement highlight the necessity of studying convergence properties, especially absolute convergence, in the context of infinite series. This article aims to provide a comprehensive understanding of these concepts, contributing to a deeper appreciation of the complexities involved in the world of infinite series.