Euler and the Cubic Basel Problem. Issue 4 (12th March 2021)

• Record Type: Journal Article
• Authors: Dunham, William
• Is Part Of: American Mathematical Monthly. Volume 128:Issue 4(2021)
• Page Start: 291
• Page End: 301
• Publication Date: 2021-03-12
• Subjects: MSC: Primary 01A50; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2021.1865014
• Languages: English
• ISSNs: 0002-9890

Abstract

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.