Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

I would prefer avoid using Taylor expansion...

share|cite|improve this question
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
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.