Friday, 24 May 2013

sequences and series - How to show that the limit limntoinftysinfracntheta2sinfrac(n+1)theta2 doesn't exist?



I've been trying to show that nsinnθ diverges (for most thetas at least), and I've come up with this expression for the partial sum (up to a multiplicative constant), and now I want to show that its limit doesn't exist:




lim



But I don't know how to proceed with this. Both terms are divergent, but that doesn't mean their product necessarily diverges (though in this case it sure seems so). Is there a straightforward way to show this limit doesn't exist?



EDIT: I want to clarify that while I did originally set out to show the divergence of a series, that's not the aim of this question, which is how to rigorously show a limit doesn't exist. I can show that the limit doesn't equal 0, but I want to learn how to show that it can't equal any other number as well.


Answer



Note that \sin\alpha\sin\beta=\frac12(\cos(\alpha-\beta)-\cos(\alpha+\beta)), hence
\sin\frac{n\theta}2\sin\frac{(n+1)\theta}2=\frac12\left(\cos\frac\theta2-\cos\bigl((n+\tfrac12)\theta\bigr)\right)
so unless \theta is special ...


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