About Uniform Convergence of $\sum_{n=1}^\infty\frac{\sin nx}{n}$ on $[0,2\pi]$

Question

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.

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

1

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.

edited Apr 13 '17 at 12:20

Community

answered May 22 '13 at 14:28

Mhenni Benghorbal

3

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 $\pi$. – David Mitra May 22 '13 at 14:57
@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. – Mhenni Benghorbal May 22 '13 at 15:16

1 Answers1