Find all $\alpha,\beta\in \mathbb R$ such that $$\sum_{n=1}^{\infty}\frac{x^{2\alpha}+n \sin x\cos\frac{2\pi n}{3}}{\sin^2x+n^2x^{2\beta}}$$ converges uniformly on $\mathbb R\setminus\{0\}.$
I would say that pointwise convergence works only for $\beta>\alpha$ and |x|>1 because of the exponential function.
EDIT: No exponential functions.