IRRACIONALIDADE DE π ao quadrado E DE POTÊNCIAS DE ε

Abstract

The central theme of this work is to present the proofs that �! and �", are irrational numbers. Through Charles Méray proof that the rational numbers were not enough to calculate the measure of the "square of the circle", it was noticed that these were not expressed by the ratio of integers and, from there, the so-called irrational numbers emerged, where the most famous in India and Greece are the numbers � and �. Given the above, we have the following problem: what can be said about the irrationality of the numbers �! and �", where � is a rational number other than zero? With this, we propose to approach some fundamental concepts and results, among which, we highlight the Fundamental Theorem of Calculus (TFC), series concepts, Convergence Tests, bringing different examples and some auxiliary lemmas. Our objective is to prove that �! and �", are irrational numbers, using exploratory, qualitative and hypothetical-deductive research as a methodology. With this, we briefly carry out a historical analysis of the irrational numbers � and �, bringing an apparatus of preliminaries on the contents that we propose to approach, concluding that the numbers �! and �", are numbers of irrational facts through the main theorems of this work.