I would really love your help with the following facts that I can't understand.
I can't understand why the following Fourier series does not converge:
$\sum_{0}^{\infty}\frac{e^{inx}}{n^2}.$
1.If I use the fact that $e^{inx}= \cos nx + i \sin nx$ so I get that the sum equals the following sum of sums: $\sum_{0}^{\infty} \frac{\cos nx}{n^2}+\sum_{0}^{\infty} \frac{\sin nx}{n^2}$, and here: why each of the sums does not converge? Can't I use Dirichlet test? $\frac{1}{n^2}\to0 , \sum_{0}^{N}|\sin nx|
2.Why does it converge for $x\in [0,2\pi]$?
Thanks a lot.