The first modern definition of the sum of a divergent series: An aspect of the rise of 20th century mathematics

In this paper my aim is to clarify the conceptual aspect of the rise of the modern concept of summability. I shall limit myself to the origin of the first modern definition of the sum of divergent series which was formulated by ERNESTO CESÀRO in 1890 after various attempts starting from 1882. The historical circumstances of these attempts, closely related to the rise of 20th century axiomatics, are a very interesting example of the transition to modern mathematics. The developments following CESÀRO’s definition, above all ÉMILE BOREL’s contributions, which actually gave rise to summability theory, are beyond the scope of this article. In the first part, I outline the conceptions underlying the different solutions of the problem of the sum of divergent series and point out that the modern use of the term ’definition’ differs from that of 18th and 19th century. I thus highlight that CESÀRO’s real novelty was precisely to give the first modern definition of the sum and not simply the first definition, which is actually due to EULER at the middle of 18th century. In the second part, I briefly illustrate the two sources of CESÀRO’s definition: the permanence of the formal method during the 19th century and the consequences of the interpretation of CAUCHY’s rigourous style by epsilontics in the 1880s (inter alia I point out that one of HÖLDER’s theorems was later incorrectly conceived as defining the sum of divergent series). In the third part, I present in specific and selected detail the evolution of CESÀRO’s thought from 1880 to 1890. The acknowledgement of the usefulness of divergent series, the attempts to “establish the fundamental principles of an asymptotic theory of numbers”, and the developments of the foundations of mathematics led him to different approaches to the problem of their sum.