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Thread: An Inequality?

  1. #1
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    An Inequality? (A Challenging One: Prove or disprove)

    Prove or disproe:
    For any positive integer n and $\displaystyle x \in [0,\pi], \quad \sum_{k=1}^n \frac{1}{k} \sin kx \geq 0$
    Last edited by elim; Mar 4th 2010 at 07:50 PM. Reason: Make the title more specific
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    Quote Originally Posted by elim View Post
    Prove or disproe:
    For any positive integer n and $\displaystyle x \in [0,\pi], \quad \sum_{k=1}^n \frac{1}{k} \sin kx \geq 0$
    Dear elim,

    You could show that this statement is correct using mathamatical induction. Can you give it a try? If you need more help please don't hesitate to reply back.
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  3. #3
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    Some how I just could not see this clearly before

    Say n (>1) is the smallest integer such that for some minimal point$\displaystyle x_0 \in (0,\pi)$,
    $\displaystyle \sum_{k=1}^n \frac{1}{k} \sin kx_0 \leq 0$,
    then $\displaystyle \sin nx_0 < 0$

    On the other hand the critical points for the function $\displaystyle f_n(x) = \sum_{k=1}^n \frac{1}{k}\sin kx$ are
    $\displaystyle x = \frac{(2m+1)\pi}{n+1}, \quad m=0,\cdots,\left[\frac{n-1}{2}\right]$
    They are the points in $\displaystyle (0,\pi)$ such that
    $\displaystyle \sum_{k=1}^n \cos kx = \frac{\cos \frac{(n+1)x}{2} \sin \frac{nx}{2}}{\sin \frac{x}{2}} = 0$
    (note that $\displaystyle \sin \frac{nx_0}{2} \neq 0$ by our finding above)
    But $\displaystyle \sin \frac{n(2m+1)\pi}{n+1} = \sin \left( 2m\pi+\frac{n-2m}{n+1}\right) = \sin \frac{n-2m}{n+1} > 0$

    Thus we get a contradiction and we actually proved that
    $\displaystyle \forall n \in \mathbf{N}^+, \forall x\in (0,\pi) \left( \sum_{k=1}^n \frac{1}{k}\sin kx > 0\right)$

    This is a bit stronger statement.

    Thanks Sudharaka for the direction. I thought this problem needs more advanced tech to solve.
    Last edited by elim; Mar 5th 2010 at 08:23 AM.
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    Quote Originally Posted by elim View Post
    It seems that I still not be able to figure out the way of using induction.
    say n (>1) is the smallest integer such that for some $\displaystyle x_0 \in (0,\pi)$,
    we have $\displaystyle \sum_{k=1}^n \frac{1}{k} \sin kx \leq 0$,
    then what?

    I did get the critical points for the function $\displaystyle f_n(x) = \sum_{k=1}^n \frac{1}{k}\sin kx$.
    They are $\displaystyle x = \frac{(2m+1)\pi}{n+1}, \quad m=0,\cdots,\left[\frac{n-1}{2}\right]$
    They are the points in $\displaystyle (0,\pi)$ such that
    $\displaystyle \sum_{k=1}^n \cos kx = \frac{\cos \frac{(n+1)x}{2} \sin \frac{nx}{2}}{\sin \frac{x}{2}} = 0$
    Dear elim,

    We have to show that for any positive integer n and $\displaystyle
    x \in [0,\pi], \quad \sum_{k=1}^n \frac{1}{k} \sin kx \geq 0$

    When n=1,

    $\displaystyle \sum_{k=1}^1 \frac{1}{k} \sin kx =sinx\geq{0}~Since,~x \in [0,\pi]$

    Therefore the expression is true for n=1

    Suppose the expression is true for n=p where $\displaystyle p\in{Z^{+}}$

    Then, $\displaystyle \sum_{k=1}^p \frac{1}{k} \sin kx\geq0$

    Now, $\displaystyle \frac{1}{(p+1)} \sin (p+1)x\geq{0}~Since,~p+1>0~and~x\in[0,\pi]$

    Therefore, $\displaystyle \sum_{k=1}^p \frac{1}{k} \sin kx+\frac{1}{(p+1)} \sin (p+1)x\geq{0}$

    Hence, $\displaystyle \sum_{k=1}^{p+1} \frac{1}{k} \sin kx\geq0$

    Therefore our the expression is true for $\displaystyle n=p+1~where~p\in{Z^+}$

    Hence by mathamatical induction the expression is true for all $\displaystyle n\in{Z^+}$

    Hope this will help you.
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  5. #5
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    Quote Originally Posted by Sudharaka View Post
    Now, $\displaystyle \frac{1}{(p+1)} \sin (p+1)x\geq{0}~Since,~p+1>0~and~x\in[0,\pi]$
    This is not true. And is the hard part.

    On the other hand, actually I did used math induction to prove the inequality is true for all positive integer n and all $\displaystyle x \in [0,\pi]$ by showing that no such an integer n>1 and $\displaystyle x_0 \in (0,\pi)$ to let
    $\displaystyle \sum_{k=1}^n \frac{1}{k} \sin kx_0 \leq 0$ happen assuming

    (**) $\displaystyle \sum_{k=1}^m \frac{1}{k} \sin kx_0 > 0$ for $\displaystyle 1 \leq m < n, \quad x \in (0,\pi)$ which is true for m=1
    Last edited by elim; Mar 6th 2010 at 09:33 AM.
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