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Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$.

The problem is that I do not know for what function is $f$ converging to. I think that maybe complex Fourier series might help here, but I'm not completely confortable with that tool.

Any help?

u1571372
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1 Answers1

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HINT: In order to establish whether the series converges or not, you do not need to actually know its limit. You are only asked to establish the existence of the limit, so you can simply apply one of the Convergence Tests, e.g. Dirichlet test.

PS Before applying these tests, it might be convenient to recall the Euler's Formula and split you series into two:

$$ e^{ix} = \cos x + i\sin x \implies f(x) = \sum_{k=1}^{\infty} \frac{e^{ikx} }{k}= \sum_{k=1}^{\infty} \frac{\cos (kx)}{k} +i\sum_{k=1}^{\infty} \frac{\sin (kx)}{k}, $$ and then establish uniform convergence of each of the series individually.

Vlad
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  • Thank you for your help, but how would you apply Abel's Uniform Test here? I do not see a way to manipulate the given function so that all the conditions in the test are checked... – u1571372 May 18 '15 at 15:24
  • You can see how to prove the convergence of $\sum \frac{\sin(kx)}{k}$ here, for example.

    PS Apparently you can prove it even by using Dirichlet test, which is weaker than Abel's

     – Vlad May 23 '15 at 01:34
  • But i need uniform convergence, not only pointwise convergence like in the link. And how would you use abel's uniform test? For example, i don't see a good candidate for the functions $f_n$ (notice that we need $\sum_{n}^{} f_n$ to converge uniform)... – u1571372 May 25 '15 at 19:17
  • You can bound sine function by 1 and get uniform convergence. See example 5.11 here – Vlad May 25 '15 at 23:43