Summary families

The law of addition on scalars or vectors verifies certain properties, such as associativity and commutativity, which allow us to naturally define sums of finite families, to which the properties of addition, considered as a binary operation, are easily extended. Difficulties arise when we consider extending summation to discrete infinite families, the archetype of which is the sequence indexed by $\mathbb{N}$ . If this summation is to be convenient to use, it must retain the properties of finite summation: associativity and commutativity, for example. Such conservation is possible at the cost of a certain limitation of the families studied.

To do this, we first study positive families, for which there is only an accumulation phenomenon, without compensation, which in all cases allows us to attribute a finite or infinite sum to the family. We then limit ourselves to the study of scalar or vector families whose moduli (or norms) have a finite sum. It is then not difficult to assign to the initial family a scalar (or vector) sum that possesses all the desirable virtues: this is the theory of summable families, which is the subject of this article.

When the family cannot be summed up, we need to develop an ad hoc theory that takes into account the original problem. Very often, the way in which the family is indexed provides an indication. If equidistant electric charges are arranged on a half-line, nothing is more natural than to number them, i.e. to index them by $\mathbb{N}$ . If we consider the potential developed at a point, for example the origin of the half-right, we can first consider the sum of the potentials corresponding to the first n charges, then examine the limit of these sums when n tends towards infinity. This approach leads to the theory of series, discussed in the article "Summatory processes" [AF 73].

When this procedure diverges (i.e. fails to assign a sum to the sequence), we can study the sequence of arithmetic averages of the terms in the sequence (Cesaro's procedure), or introduce the generating integer series associated with the sequence, then examine the limit of this integer series when the variable tends towards 1. We then obtain Abel's summation procedure.

These and similar procedures have a certain consistency, in that when two procedures converge, the sums allocated will be equal, even when the "sum" is $+\mathrm{\hspace{0.17em}}\infty$...