Divergent series have appeared in mathematics for a long time. They were used widely throughout the 17th and 18th centuries, sometimes with great profit and sometimes without. Many mathematicians who used them were frequently in doubt about their validity while others seemed totally unconcerned. Throughout the period, controversy surrounded the use of divergent series mainly because contradictions were often derived through their application. One of the earliest discussions regarding a divergent series occurred when the infinite series

The development of the theory of summable divergent series from 1880 to 1925

I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

Taking mathematical fictions seriously

Taking into account the renewed interest in the applications of the theory of divergent series, we consider it proper to revaluate this little known part of Borel's works. We begin With the analysis of the circumstances and main sources which support Borel's early research. Then we briefiy describe how Borel built his theory. Finally, making use of diverse and original sources, we intend to clarify the mathematical and epistemological importance of Borel's theory and its role in the rational modification of mathematical practice.

https://dialnet.unirioja.es/descarga/articulo/62137.pdf

Revalorización de la obra temprana de Emile Borel sobre sumación de series divergentes

Teoría de los algoritmos lineales de convergencia y de sumación

Über einen Grenzwertsatz aus der Theorie der unendlichen Folgen

History and synopsis of the theory of summable infinite processes

Formulas for Reducing a Quadratic Form to a Sum of Squares

Three Books on Infinite Series@@@Theorie und Anwendungen der Unendlichen Reihen.@@@Elements of the Theory of Infinite Processes.@@@An Introduction to the Operations with Series.

Tomlinson Fort

Papers

History and synopsis of the theory of summable infinite processes

Formulas for Reducing a Quadratic Form to a Sum of Squares

Three Books on Infinite Series@@@Theorie und Anwendungen der Unendlichen Reihen.@@@Elements of the Theory of Infinite Processes.@@@An Introduction to the Operations with Series.