TITLE: $\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?
QUESTION [5 upvotes]: Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions.

Question 1 : Is the following true?
  $$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$
Question 2 : Is the following true?
  $$\sum_{k=1}^{n}\frac{\sin(kx)}{k}\gt0$$

Motivation : I've known the smaller $n$ cases. These seems true by using computer. I would like to know not only whether there are true but also how to prove that. Can you help?

REPLY [2 votes]: Question 1 is Young inequality(1912);can see 
http://sci-prew.inf.ua/v134/1/S0305004102006357.pdf
and can see Young's paper(1912),But I can't download:http://plms.oxfordjournals.org/content/s2-12/1/41.extract