Solution of the Basel problem using the Feynman integral trick

Abstract

Euler's solution in 1734 of the Basel problem, which asks for a closed form expression for the sum of the reciprocals of all perfect squares, is one of the most celebrated results of mathematical analysis. In the modern era, numerous proofs of it have been produced, each emphasizing a different style of calculation, as a way of testing the power of some demonstration method. It's often thought that solutions using calculus need to involve advanced contour integration techniques or geometric coordinate transformations. We show that this is not the case, as the result can be derived by analyzing basic properties of a particular one-dimensional integral, and as such, can be obtained with techniques typical of regular calculus tests and math competitions.