55. (USS 2) Find all \( x \) for which for all \( n \),
\[ \sin x + \sin 2x + \sin 3x + \cdots + \sin nx \leq \frac{\sqrt{3}}{2} \]
55. It is enough to find all $x$ from $(0,2 \pi]$ such that the given inequality holds for all $n$. Suppose $0 < x < 2 \pi / 3$. We have $\sin x > \sqrt{3} / 2$. Suppose now that $2 \pi / 3 \leq x < 2 \pi$. We have  $$ \sin x + \cdots + \sin n x = \frac{\cos \frac{x}{2} - \cos \frac{2 n + 1}{2} x}{2 \sin \frac{x}{2}} \leq \frac{\cos \frac{x}{2} + 1}{2 \sin \frac{x}{2}} = \frac{\cot \frac{x}{4}}{2} \leq \frac{\sqrt{3}}{2} . $$  For $x = 2 \pi$ the given inequality clearly holds for all $n$. Hence, the inequality holds for all $n$ if and only if $2 \pi / 3 + 2 k \pi \leq x \leq 2 \pi + 2 k \pi$ for some integer $k$.