What does the series $\sum\limits_{n=-\infty}^{\infty} \frac{1}{n}e^{i \pi n x}$ ($n\neq 0)$ converges to on $(-1,1)$?

Question

What does the series $\sum\limits_{n=-\infty}^{\infty} \frac{1}{n}e^{i \pi n x}$ ($n\neq 0)$ converges to on $(-1,1)$? I tired plotting this series for different values of $n$ and it seemed it is zero everywhere, which I know it cannot be true. Moreover, I know it converges because $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$ is finite.