Convergence of $\sum_{1}^{\infty}(-1)^n\frac{\sin(n\theta)}{n}$

Question

How can we prove the convergence of following series, for each $\theta$, by elementary means? $$\sum_{1}^{\infty}(-1)^n\frac{\sin(n\theta)}{n}$$

The series appears as the Fourier series for $f(\theta)=\theta$ considered over $[-\pi,\pi]$.