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

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