Uma jornada fascinante ao mundo de PI: a série de Leibniz e a conexão inesperada com os números ímpares

Abstract

This dissertation investigates the number pi through three interrelated approaches: mathematical, historical, and pedagogical. The study focuses on the Leibniz series, presenting three distinct proofs: a geometric approach, using arc approximations via similar triangles; Leibniz's Transmutation Theorem, which transforms integrals to derive the series; and the Taylor series expansion of the arctangent function. Furthermore, the connection between pi and odd numbers is explored, including infinite products involving odd primes. The research also develops a teaching sequence for high school, translating these advanced concepts into accessible activities, with an emphasis on geometric visualization and mathematical investigation. Methodologically, it combines a literature review (primary and secondary sources) with the creation of educational materials tested in a school environment. The results stand out in three aspects: the unification of proofs of the Leibniz series, the relationships between pi and odd numbers, and the preliminary validation of the pedagogical approach. It is concluded that advanced topics, such as infinite series, can be introduced intuitively into basic education, enriching students' mathematical training.