Expansion of the infinite product $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ etc. into a simple series

Abstract

Translated from the Latin original "Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number theorem. He gives his original proof explained a bit differently, and then gives a different proof. However this second proof is still rather close to his original proof. To understand the two proofs, I wrote them out using subscript notation and sum/product notation. It would be a useful exercise to try to really understand the proofs without using any modern notation. The right notation takes care of a lot for us, which we would otherwise have to keep active in our minds.