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
)(1
x
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.