This question already has an answer here:

Is $\sum_{n=1}^\infty\dfrac{\sin nx}{n}$ uniform convergent on $[0,2\pi]$?

I think it is not. However, I could not prove it by Cauchy's criterion.


marked as duplicate by Start wearing purple, David Mitra, Micah, Julian Kuelshammer, user33321 May 22 '13 at 15:17

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


A related problem.

Hint: Notice that the series is the Fourier series of the function

$$ -\frac{\pi+x}{2}. $$

Now, use the following result:

Theorem: The Fourier series of a $2\pi$-periodic continuous and piecewise smooth function converges uniformly.

  • 2
    $\begingroup$ It's the Fourier series of the periodic extension of your function over $[0,2\pi]$. This is not continuous at even integer multiples of $\pi$. $\endgroup$ – David Mitra May 22 '13 at 14:57
  • $\begingroup$ @DavidMitra: We know it is the periodic extension. I just gave him a hint to think and reach a conclusion. Read the comments under this post. $\endgroup$ – Mhenni Benghorbal May 22 '13 at 15:16

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