Uniform convergence of function series

[twoflower]

Hi,

I've little probles with this one:

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

$$\sum_{k = 1}^{\infty} \frac{\cos kx}{k}$$

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

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

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

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

So it could converge on $x \in [2n\pi + \epsilon, 2(n+1)\pi - \epsilon], \mbox{where } \epsilon \in (0, \pi)$ 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.

 

[fourier jr]

$$a_k = \frac{1}{k} = \frac{1}{L} \int_{-\pi}^{\pi} f(x)cos(\frac{n \pi x}{L})dx$$

so obviously $$L = \pi$$ & 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.

 

[twoflower]

$$a_k = \frac{1}{k} = \frac{1}{L} \int_{-\pi}^{\pi} f(x)cos(\frac{n \pi x}{L})dx$$

so obviously $$L = \pi$$ & 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.

 

[Galileo]

Dirichlet's criterium will work fine:

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

$$\sum_{n=0}^{\infty} a_nb_n$$ converges.


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

Check the following series:
$$\sum_{n=0}^{\infty} z^n$$
for $z\in \mathbb{C}, |z|=1, z\not=1$. Show that the partial sums are bounded and note that $\cos kx = \Re e^{ikx}$.

 

[saltydog]

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.

 

[Galileo]

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?

 

[saltydog]

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.

 

[twoflower]

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).

 

[saltydog]

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:

$$\sum_{n=1}^{\infty}\frac{Cos(n)}{n}$$

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:

$$\frac{2}{|1-e^{ix}|}$$

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