Euler and His Work on Infinite Series

Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. It was also said (by Laplace) that all mathematicians were his students. It is appropriate in this, the tercentennial year of his birth, to revisit him and survey his work, its offshoots, and the remarkable vitality of his themes which are still flourishing, and to immerse ourselves once again in the universe of ideas that he has created. This is not a task for a single individual, and appropriately enough, a number of mathematicians are attempting to do this and present a picture of his work and its modern resonances to the general mathematical community. To be honest, such a project is Himalayan in its scope, and it is impossible to do full justice to it. In the following pages I shall try to make a very small contribution to this project, discussing in a sketchy manner Euler’s work on infinite series and its modern outgrowths. My aim is to acquaint the generic mathematician with