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I've got some questions to the following task.

Let [itex]r\in\mathbb{Q}[/itex]. Determine the partial limits of the complex sequence [itex](z_n)[/itex] defined by [itex]z_n=e^{i2\pi{rn}}[/itex].

There is also a hint given: Set [itex]r=\frac{p}{q}[/itex] with [itex]p\in\mathbb{Z},q\in\mathbb{N}\setminus{\{0\}[/itex], so that the fraction [itex]\frac{p}{q}[/itex] is irreducible. From [itex]\frac{p}{q}\alpha\in\mathbb{Z},\alpha\in\mathbb{Z}[/itex] then follows [itex]\frac{\alpha}{q}\in\mathbb{Z}[/itex].

Every complex number is a point of the complex plane and can be written as:

[itex]cos(t)+i\\sin(t)=e^{it}[/itex], so the defined sequence elements can be rewritten as: [itex]z_n={e^{i2\pi{rn}}=cos(n\cdot{2\pi\frac{p}{q}})+i\\sin(n\cdot{2\pi\frac{p}{q})=(cos({2\pi\frac{p}{q}})+i\\sin(2\pi\frac{p}{q}))^n[/itex]

From the equation it's pretty clear that we're dealing with the q-th roots of 1 in the complex, and that p<q, because we have q points in the complex plane, whose radius-vectors are at angle fewer or equals than 2*pi*p. If we add the factor n, we increase the angle of each radius-vector of a particular point (with given p/q) n-fold. Is it right?

Now, I don't know at the moment how to get to the partial sequences. I can't see how the hint could help. It says that alpha is greater-equals q, but why should it be so in our case (I mean q=3/4 for example, then alpha could still be 1,2,3, couldn't it?...we would still get some points)? If it must in fact be greater-equals q and alpha/q is an integer, then does it mean that we could take all elements with indices, which are multiple of q, to get a subsequence?

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# Homework Help: Sequence on a circle

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