On sums of numerical series and Fourier series1

¹Dedicated to Prof. J.F.C.A. Meyer on his 60th birthday.

Abstract

We discuss a class of trigonometric functions whose corresponding Fourier series, on a conveniently choosen interval, can be used to calculate several numerical series. Particular cases are presented and two recent results involving numerical series are recovered.

¹Dedicated to Prof. J.F.C.A. Meyer on his 60th birthday.

Keywords:

• numerical series
• Fourier series
• sum of series

Acknowledgements

We are grateful to the anonymous referee, who made several important remarks on Fourier series. E.C.O is also grateful to Fapesp (06/52475-8) for a research grant and to Prof. J. Vaz Jr and Dr J. Emílio Maiorino for many useful discussions.

Notes

¹Dedicated to Prof. J.F.C.A. Meyer on his 60th birthday.

Note

1. Suppose the trigonometric series to be convergent. The necessary and sufficient condition for this series to be a Fourier series is that or, in this case, be a convergent series. A trigonometric series as the one above is a Fourier series if it is term-by-term integrable. In this case, in particular, it converges uniformly on the interval to a function f (x). We may then multiply it scalarly by and we obtain , which are the so-called Fourier coefficients of f (x).