Evolutio producti infiniti (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem

TL;DR

This paper explores the expansion of an infinite product series related to pentagonal numbers, deriving the pentagonal number theorem and discussing its application to partition numbers, including a translation of Euler's original work.

Contribution

It provides a detailed derivation of the pentagonal number theorem from the infinite product series and offers a translation of Euler's original Latin text.

Findings

Derivation of the pentagonal number theorem from the series expansion

Connection between the series and partition numbers

Historical translation of Euler's original work

Abstract

This paper does exactly what the title says it does. It expands the given series to arrive at the familiar "pentagonal number" expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Euler's Latin original into German.