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

Abstract

Summary If we consider the Madhava–Gregory–Leibniz series π 4 = 1 − 1 3 + 1 5 − ⋯ , which is commonly referred to as the Gregory series, and then compare it to Euler’s formula π 2 8 = 1 + 1 3 2 + 1 5 2 + ⋯ , 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 ζ ( 2 ) = π 2 / 6 , based on the idea of squaring 1 − 1 3 + 1 5 − ⋯ = π 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 ζ ( 2 ) = π 2 / 6 that also relies on squaring Gregory’s series.