无穷乘积$(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$等展开为简单级数
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
摘要：
翻译自拉丁原文“Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem”（1775）。Enestroem指数中的E541。在本文中，欧拉是重温他的证明五边形数定理。他给出了他的原始证明，解释有点不同，然后给出了一个不同的证明。然而，这第二个证明仍然是相当接近他原来的证明。为了理解这两个证明，我用下标表示法和和积表示法把它们写出来。这将是一个有用的练习，试图真正理解证明，而不使用任何现代符号。正确的符号为我们提供了很多帮助，否则我们必须在头脑中保持活跃。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
