Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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 Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniform convergence of function series

**Physics Forums | Science Articles, Homework Help, Discussion**