TITLE: Example of a trigonometric series that is not fourier series?
QUESTION [8 upvotes]: My textbook doesn't give any example of this kind of series. Could you provide some?
Trigonometric series is defined in wikipedia as :
$A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$
When 
$A_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x) \cos{nx} dx\qquad (n=0,1,2,3 \dots)$
$B_{n}=\frac{1}{\pi} \int^{2 \pi}_0 f(x) \sin{nx} dx\qquad (n=1,2,3, \dots)$
It is fourier series.
thanks.

REPLY [2 votes]: Theorem. If $a_n>0$ and $\sum_{n>0}\frac{a_n}{n}=\infty$. Then $\sum_1^\infty a_n\sin nt$ is not a Fourier series. (AN INTRODUCTION TO HARMONIC ANALYSIS, Yitzhak Katznelson)