Hermes Antonio PEDROSO[1]

§    ABSTRACT: The aim of this works to discuss some results obtained first time by Euler in 1736 about the sums of series of the type , k Î Z+. We first make a brief of the Scientific activities of the 18 th century including the work and life of Euler . In the sequel we present Euler’s curious procedure to get the sums of , k Î Z+.. Finally, we show how to get those sums, from a rigorous point of view, by using Fourier series, simple and double integrals and power series. We further leave for the interested reader some results concerning the subject and a challenge: since the time Euler, with all modern techniques, few facts are know about the sums of the series , k Î Z+.. For instance, only in 1978 it was shown by the french mathematician R. Apéry that the sums , k Î Z+. is irrational number, and nothing else is know about that number. For the sum even now we don`t known if it is a rational or an irrational number. So, in a surprising way, one gets to the boundary of the knowledge concerning those questions.

§    KEYWORDS: Euler sums; 18 th century; Fourier series; integrals; power series.



[1] Departamento de Matemática - Instituto de Biociências , Letras e Ciências Exatas - UNESP - São José do Rio Preto - SP - Brasil.