$\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

Question

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?

score 2 · Accepted Answer · answered Oct 25 '13 at 15:59

2

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

answered Oct 25 '13 at 15:59

math110

93,304

$\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

1 Answers1