Abstract

In 1734, Leonhard Euler summed the infinite series of reciprocals of the squares, thereby solving a challenge known as the “Basel problem.” He later extended his method to find closed-form sums for the reciprocals of 4^th, 6^th, and other even powers. But those techniques did not yield a value for the sum of the reciprocals of the cubes. Here, we show how Euler tried to evaluate this series by transforming it into the sum of a strange constant and an even stranger integral.

中文翻译：

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欧拉和三次巴塞尔问题

摘要

1734年，莱昂哈德·欧拉（Leonhard Euler）对平方的倒数进行了无穷级的求和，从而解决了一个被称为“巴塞尔问题”的挑战。后来，他扩大了他的方法找到封闭形式的总和为4的倒数^日，6^日，和其他甚至权力。但是那些技术并没有得出立方体的倒数之和的值。在这里，我们展示了欧拉如何尝试通过将其转化为一个奇怪的常数和一个甚至更陌生的积分之和来评估该系列。