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 $$\forall n \in \mathbf{N}^+, \forall x\in (0,\pi) \left(\displaystyle{\sum_{k=1}^n \frac {1}{k} \sin kx > 0} \right)$$
03-04-2010, 04:41 PM (This post was last modified: 03-05-2010 11:41 AM by elim.)
Post: #1
 elim Moderator Posts: 544 Joined: Feb 2010 Reputation: 0
$$\forall n \in \mathbf{N}^+, \forall x\in (0,\pi) \left(\displaystyle{\sum_{k=1}^n \frac {1}{k} \sin kx > 0} \right)$$
Show that [试证] $$\forall n \in \mathbf{N}^+, \forall x\in (0,\pi) \left(\displaystyle{\sum_{k=1}^n \frac {1}{k} \sin kx > 0} \right)$$
03-05-2010, 12:25 PM (This post was last modified: 03-05-2010 12:26 PM by elim.)
Post: #2
 elim Moderator Posts: 544 Joined: Feb 2010 Reputation: 0
The proof of the inequality
Let $$n$$ be the smallest positive integer such that for some minimal point $$x_0 \in (0,\pi)$$, $$\displaystyle{\sum_{k=1}^{n} \frac{1}{k}\sin kx_0 \leq 0}$$
Then clearly $$n > 1$$ and $$\sin n x_0 < 0$$
But $$\displaystyle{\frac{d}{dx} \sum_{k=1}^{n}\frac{1}{k}\sin kx = \sum_{k=1}^{n} \cos kx = \frac{\cos \frac{(n+1)x}{2} \sin \frac{nx}{2}}{\sin \frac{x}{2}}}$$ and $$\displaystyle{\left(\sin \frac{nx}{2} = 0 \right)\Rightarrow (\sin nx = 0)}$$,
we see that $$\cos \frac{(n+1)x_0}{2} = 0$$ and so for some integer $$m$$,
$$(n+1)x_0/2 = (m+1/2)\pi \Rightarrow x_0 \in \left\{\displaystyle{\frac{(2m+1)\pi}{n+1}: m = 0,\cdots,\left[\frac{n-1}{2} \right]}\right \}$$ since $$x_0 \in (0,\pi)$$
This means that for some $$m$$ we have $$x_0 = (2m+1)\pi /(n+1)$$ hence $$\sin nx_0 = \sin \displaystyle{\left(2m\pi+\frac{(n-2m)\pi}{n+1} \right) = \sin \frac {(n-2m)\pi}{n+1}} > 0$$