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

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
@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

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.
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