A prova de Euler para a resolução do problema da Basileia

Abstract

From the development of new tools in mathematics, the studies of infinite series constituted themselves in a high level in the scenario of pure and applied mathematics. The present work approaches a historical contextualization of the life and work of Leonhard Paul Euler, presenting a study about Euler’s proof for solving the Basel problem. This problem, which remained unsolved for a long time, consisted of formulating the result of the sum of the inverses of the perfect squares. In general, the research aims to analyze Euler’s ingenious and elucidative proof until the validation of the proof by the Weierstrass factorization theorem. It also approaches other elementary resolutions and some rigorous evidences for solving the problem. It discusses yet suggestions for activities related to the sum of infinite series in high school. Thus, it is expected that this work may contribute significantly to the teaching-learning of teachers and students in the academic scope and in high school.