On the Basel Problem and the Square of Gregory’s Series

Summary

If we consider the Madhava–Gregory–Leibniz series

$$\begin{array}{c}\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\cdots ,\hfill \end{array}$$

which is commonly referred to as the Gregory series, and then compare it to Euler’s formula

$$\begin{array}{c}\frac{{\pi }^2}{8}=1+\frac{1}{3^2}+\frac{1}{5^2}+\cdots ,\hfill \end{array}$$

the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler’s formula, or its equivalent formulation $\zeta (2)={\pi }^2/6$, based on the idea of squaring $1-\frac{1}{3}+\frac{1}{5}-\cdots =\frac{\pi }{4}$, including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of $\zeta (2)={\pi }^2/6$ that also relies on squaring Gregory’s series.

MSC:

• 11M99

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

John M. Campbell

John M. Campbell (MR Author ID: 1079559) completed his Ph.D. in pure mathematics at York University, and graduated first class with distinction with a Specialized Honours B.Sc. degree in mathematics also from York University. He has been awarded the prestigious Carswell Scholarship and the Irvine R. Pounder Award, and has worked as a research assistant at York University and at the Fields Institute for Research in Mathematical Sciences.

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Paul Levrie

Paul Levrie (MR Author ID: 230199) is Professor of Mathematics at the Faculty of Applied Engineering of the University of Antwerp, Belgium. He obtained his Ph.D. in mathematics at KU Leuven in 1987 in the field of numerical analysis. Since then, he has been busy teaching and trying to find easy ways to explain difficult concepts, or elementary proofs for known mathematical results. In an attempt to make mathematics more popular in Flanders, Belgium, he coauthors a blog (in Dutch) entitled Wiskunde is sexy (Mathematics is sexy), and gives popularizing talks about mathematics. The blog led to the publication of a book about prime numbers for the interested (non-)mathematician. He is one of a team of people organizing the monthly MathsJam in Antwerp.

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