In some exercises for an analysis course, we are asked to prove that the function series $\sum f_n$ converges uniformly on $[-1,1]$, where
$$ f_n:[-1,1]\to \mathbb{R}:x \mapsto \frac{x^n sin(nx)}{n} $$ Now on $[-1/2,1/2]$ one can use the Weierstrass M-test rather easily. On $[1/2,1]$ one can use the Dirichlet criterion after realizing that $$ \vert \sum_{k=1}^n sin(kx) \vert \leq \frac{1}{sin(\frac{x}{2})}. $$ But this does not seem to work for $[-1,-1/2]$, because we do not have montonicity there... Can anyone help me out in this region?