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:




limnsinnθ2sin(n+1)θ2



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 sinαsinβ=12(cos(αβ)cos(α+β)), hence
sinnθ2sin(n+1)θ2=12(cosθ2cos((n+12)θ))


so unless θ is special ...


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