# Jean le Rond d’Alembert: The Concept of Limits and Infinite Series

June 27, 2021Defining the differential as the limit of a function is one of the cornerstones of mathematics. The only mathematician to defend this concept in his day was d’Alembert. However, he, too, was deeply attached to the traditions that regarded geometry as the mother of all sciences and believed that all results in mathematics should be expressed through geometry; therefore, he could never formalize the concept of limit and impose it on others. However, many of d’Alembert’s results in mathematics are closely related to his understanding of limits.

D’Alembert is considered first a mathematician and then a physicist. While this is generally true, d’Alembert, in his own words, often arrived at mathematical conclusions by physical instinct. D’Alembert was never drawn into the abstract world of theoretical mathematics, continuing the tradition of Descartes. Nor did he believe that theoretical mathematics should reduce everything to a set of algorithms.

The concept of limits led d’Alembert to develop rules for operating infinite series. In volume 5 of his Opuscules (“Booklets”), he developed a method for finding the convergence or divergence of infinite series. According to this:

The R number is calculated. As a result, for R>1 the series is divergent, while R<1 is convergent. However, if R=1, no decision can be made for convergence or divergence.

Besides his original contributions to mathematics, d’Alembert’s main aim was the description of physical phenomena through mathematics. According to D’Alembert, first the differential equations determining the state would be written, and then they would be integrated. For this, mathematical physicists had to develop new methods. D’Alembert’s important contributions such as obtaining the wave equation and finding its solution always followed this sequence.

Prepared by: Sociologist Ömer Yıldırım

Source: Atatürk University Sociology Department 1st Year “Introduction to Philosophy” and 2nd, 3rd, 4th Grade “History of Philosophy” Lecture Notes (Ömer YILDIRIM)