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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.

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

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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.

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    $\begingroup$ 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

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