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I have this assignnent

Show that $$\left|\frac{sin(\frac{nx}{2})} {sin(\frac{x}{2})}\right| \leq n \space (x \ne 0,\pm2\pi, \pm4\pi, ...)$$

The question comes with an hint

Argue first that $z = e^{ix}$

Then the left-side equals $ |\frac{1 - z^n} {1 - z}| $

I have managed to solve other questions in the assignment but this one has been problematic.

Any help would be appreciated.

Thanks in Advance.

1 Answers 1

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Observe \begin{align} \left|1+e^{i\theta}+e^{i2\theta}+\ldots + e^{i(n-1)\theta}\right|= \left|\frac{1-e^{in\theta}}{1-e^{i\theta}}\right| = \left|\frac{e^{in\theta/2}}{e^{i\theta/2}}\frac{\sin \frac{n\theta}{2}}{\sin\frac{\theta}{2}}\right|= \left|\frac{\sin \frac{n\theta}{2}}{\sin\frac{\theta}{2}}\right| \end{align} which means \begin{align} \left|\frac{\sin \frac{n\theta}{2}}{\sin\frac{\theta}{2}}\right| \leq 1+|e^{i\theta}|+\ldots+|e^{i(n-1)\theta}| =n. \end{align}

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    Thanks a bunch ... This is very helpful but I dont know where the first iidentities are coming from.. – 2017-02-01
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    the first identity is the partial sum of a geometric series. – 2017-02-01
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    Thanks a lot. U re my hero – 2017-02-01
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    @JackyChong {+1} This was awesome. I had this in mind: $$\left|\frac{\sin \left(\frac{nx}{2}\right)}{\sin \left(\frac{x}{2}\right)}\right| |\sin \left(a+\frac{n-1}{2}x\right)| = | \sum_{k=0}^{n-1} \sin{\left(a+kx\right)} | \leq \sum_{k=0}^{n-1} | \sin{\left(a+kx\right)} | \leq n$$ and wasn't able to come up with anything further. – 2017-02-01