On the convergence of some alternating series

We establish a necessary and sufficient condition for the convergence of the series $sum_{n=1}^{infty} (-1)^{n}|sin(pi nx)|n^{-theta}$ in terms of the rational approximations to x. In particular, it follows from our results that the series $sum_{n=1}^{infty} (-1)^{n}|sin n|/n$ converges.