Uniform convergence of function series

In summary, the conversation is about analyzing the uniform and local uniform convergence of the series of functions \sum_{k = 1}^{\infty} \frac{\cos kx}{k}. The person is trying to solve it using Weierstrass' criterion but is having trouble proving convergence on certain intervals. Suggestions are made to use Fourier series or Dirichlet's criterion. There is also a discussion about the different names and variations of Dirichlet's criterion. Finally, a suggestion is made to use complex analysis to prove convergence.
  • #1
twoflower
368
0
Hi,

I've little probles with this one:

Analyse uniform and local uniform convergence of this series of functions:

[tex]
\sum_{k = 1}^{\infty} \frac{\cos kx}{k}
[/tex]

I'm trying to solve it using Weierstrass' criterion, ie.

[tex]
\mbox{Let } f_n \mbox{ are defined on } 0 \neq M \subset \mathbb{R}\mbox{, let }
S_n := \sup_{x \in M} \left| f_{n}(x)\right|, n \in \mathbb{N}. \mbox{ If }
\sum_{n=1}^{\infty} S_n < \infty\mbox{, then } \sum_{n=1}^{\infty} f_{n}(x) \rightrightarrows \mbox{ on } M.
[/tex]

I can see from this it won't converge for [itex]x = 2n\pi[/itex], because then

[tex]
\sup_{x \in \mathbb{R}} \left|\frac{\cos kx}{k}\right| = \frac{1}{k} =: S_{k} \mbox{ and thus } \ \sum S_k \ \mbox{diverges} \Rightarrow \sum_{n=1}^{\infty} f_{n} \mbox{ diverges}
[/tex]

So it could converge on [itex]x \in [2n\pi + \epsilon, 2(n+1)\pi - \epsilon], \mbox{where } \epsilon \in (0, \pi)[/itex] but I can't see how could I prove it, at least using this theorem. I also took a look at Abel's criterion, Dirichlet's criterion and theorems about change of sum and derivation\integral, but I didn't see the way to prove that.

Could you help me please?

Thank you.
 
Physics news on Phys.org
  • #2
[tex]a_k = \frac{1}{k} = \frac{1}{L} \int_{-\pi}^{\pi} f(x)cos(\frac{n \pi x}{L})dx[/tex]

so obviously [tex]L = \pi[/tex] & f is some even function. maybe Fourier series could help? i have no idea but when i saw that function i thought Fourier series. there's a theorem that says if f is 2-pi-periodic, continuous & piecewise-smooth then the Fourier series converges absolutely & uniformly. for that to work though you got to figure out what f is. i don't know if that makes it any easier or what.
 
Last edited:
  • #3
fourier jr said:
[tex]a_k = \frac{1}{k} = \frac{1}{L} \int_{-\pi}^{\pi} f(x)cos(\frac{n \pi x}{L})dx[/tex]

so obviously [tex]L = \pi[/tex] & f is some even function. maybe Fourier series could help? i have no idea but when i saw that function i thought Fourier series. there's a theorem that says if f is 2-pi-periodic, continuous & piecewise-smooth then the Fourier series converges absolutely & uniformly. for that to work though you got to figure out what f is. i don't know if that makes it any easier or what.

Thank you for the suggestion, Fourier jr, we've just learning Fourier series, anyway I assume the problem I posted should be able to be solved just using knowledge of uniform convergence of sequences/series of functions. I'm not still much familiar with Fourier series to try it this way.
 
  • #4
Dirichlet's criterium will work fine:

If [itex]\{a_n\}_{n=0}^\infty[/itex] is a sequence of (complex) numbers whose sequence of partial sums are bounded:
[tex]\left|\sum_{n=0}^N a_n \right| <M[/tex]
for some [itex]M \in \mathbb{R}[/itex].
And if [itex]\{b_n\}_{n=0}^\infty[/itex] is a sequence of real numbers with [itex]b_0 \geq b_1 \geq b_2 \geq ...[/itex] which tend to zero ([itex]\lim_{n \to \infty} b_n =0[/itex]) then the series:

[tex]\sum_{n=0}^{\infty} a_nb_n[/tex] converges.


So you have to show if [itex]\sum_{k=0}^N \cos (kx)[/itex] is bounded when x is not a multiple of [itex]2\pi[/itex].

Check the following series:
[tex]\sum_{n=0}^{\infty} z^n[/tex]
for [itex]z\in \mathbb{C}, |z|=1, z\not=1[/itex]. Show that the partial sums are bounded and note that [itex]\cos kx = \Re e^{ikx}[/itex].
 
Last edited:
  • #5
Nice problem. I found the thread "testing convergence of a series" in the general math forum and the link to Abel's test there helpful towards understanding it. Thanks guys.
 
  • #6
If Abel's criterion on that page (and prolly according to other people as well) is what I refer to as Dirichlet's criterion, then what is Dirichlet's criterion referring to?
 
  • #7
Galileo said:
If Abel's criterion on that page (and prolly according to other people as well) is what I refer to as Dirichlet's criterion, then what is Dirichlet's criterion referring to?

Well, I mean I found your analysis helpful too Galileo and is in fact what led me to connect the two. :smile:
 
  • #8
Galileo said:
If Abel's criterion on that page (and prolly according to other people as well) is what I refer to as Dirichlet's criterion, then what is Dirichlet's criterion referring to?

Our professor told us the Abel's test from the link as Abel's-Dirichlet's criterion, but it was bit more complex.

Now I'm looking at PDF of lecture of another professor and he titled the exactly same test as the one from the link as Dirichlet's criterion (he also has Abel's test there, it's something slightly different).
 
Last edited:
  • #9
Two flower, if I may make a suggestion as I'm learning this too, try going into the Abel's link over there and check out the example for:

[tex]\sum_{n=1}^{\infty}\frac{Cos(n)}{n}[/tex]

They use some complex analysis to prove convergence. The technique they used can be applied to your problem. For example, I found it interesting that going through the proof for your question involved considering the quotient:

[tex]\frac{2}{|1-e^{ix}|}[/tex]

Now, for what values of x does this quotient not exist?
 
Last edited:

Related to Uniform convergence of function series

1. What is uniform convergence of function series?

Uniform convergence of function series is a type of convergence in which the rate of convergence is the same at every point in the domain of the function. In other words, the function series converges to the same limit value at every point in its domain.

2. How is uniform convergence different from pointwise convergence?

Pointwise convergence refers to a type of convergence where the function series converges to a different limit value at different points in its domain. In contrast, uniform convergence means that the function series converges to the same limit value at every point in its domain.

3. What is the importance of uniform convergence in mathematics?

Uniform convergence is important because it guarantees that the limit function is continuous, which allows for the use of various mathematical tools and techniques. It is also a necessary condition for the interchange of limits and integrals, which is a crucial concept in calculus and analysis.

4. How can uniform convergence be tested?

Uniform convergence can be tested using various criteria, such as the Weierstrass M-test and the Cauchy criterion. These criteria provide conditions that, if satisfied, guarantee the uniform convergence of a function series.

5. Can a function series converge uniformly but not pointwise?

Yes, it is possible for a function series to converge uniformly but not pointwise. This can happen if the rate of convergence is different at different points in the domain, but the overall convergence is still uniform.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
494
  • Math POTW for University Students
Replies
1
Views
492
  • Calculus and Beyond Homework Help
Replies
4
Views
472
  • Topology and Analysis
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
319
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
783
  • Topology and Analysis
Replies
5
Views
1K
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
1K
Back
Top