Summary processes - The series

A summation procedure consists in assigning a "sum" to an infinite family of elements of a normed vector space. When the family is summable (cf. article of this treatise), the simplest way is to assign the sum of this summable family. When this is not the case, it is advisable to implement a summation procedure that takes account of the actual situation: this may be the indexation of the set or, again, the nature of the phenomenon being studied. For example, when decomposing a periodic phenomenon with discontinuities, it is necessary to use a Fourier series that does not correspond to a summable family: this may be a series indexed by $\mathbb{N}$ , or even the limit of symmetrical sums.

This article looks at numerical series, by far the most elementary example of a summation procedure. In addition to a few methods for studying convergence, examples are given of exact calculations of sums of such series.

In the "Asymptotic developments" article The asymptotic or numerical evaluation of sums, whether finite sums or sums of series, will be discussed. In connection with this topic, infinite products will be discussed. Other summation procedures are also covered.