SCHWEITZER 1952 Problem 10

Let $n$ be a positive integer. Prove that, for $0<x<\frac{\pi}{n+1}$ , $\sin{x}-\frac{\sin{2x}}{2}+\cdots+(-1)^{n+1}\frac{\sin{nx}}{n}-\frac{x}{2}$ is positive if $n$ is odd and negative if $n$ is even.