ABOUT CERTAIN EULER SUMS.

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.