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How do I prove convergence of $\sum\limits_{k=1}^\infty \tfrac{\sin k^2}{k}$?

I would prefer avoid using Taylor expansion...

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integral criterion.. the series will be convergent if the integral $ \int_{1}^{\infty}dx \frac{sin(x^{2})}{x} $ is convergent –  Jose Garcia Jan 31 '13 at 10:29
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@JoseGarcia No. The integral criterion only works for nonnegative, non-increasing integrands. –  Harald Hanche-Olsen Jan 31 '13 at 10:32
    
I’m pretty sure that Marvis will appear on this thread soon with a proof using Dirichlet’s Test, assuming that $ \left( \displaystyle \sum_{k=1}^{n} \sin(k^{2}) \right)_{n \in \mathbb{N}} $ is a bounded sequence. –  Haskell Curry Jan 31 '13 at 10:43

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up vote 3 down vote accepted

It is not easy. The convergence of the more general series $$ \sum_{n=1}^\infty\frac{\sin(n^k)}{n},\quad k>0, $$ is studied in this question.

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anyway.. what if the series does ont converge :) we simply truncate it and that is all, for example the series $ 1/2 + \sum_{n=1}^{\infty}cos(nx) $ does not converge –  Jose Garcia Jan 31 '13 at 11:38

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