# Applying the Weierstrass M test

1. Feb 24, 2009

### letmeknow

1. The problem statement, all variables and given/known data

$$\sum_{n=1}^{\infty}\frac{1}{n^2}cosnx$$

Apply the Weierstrass M-test to the series in order to find the domain of convergence.

2. Relevant equations

3. The attempt at a solution

Looking at the functions I think I have gottten to the following conclusion,

f1(x)=cosx <= 1

f2(x)=1/4 cos(2x) <= 1/4

I would pick my sequence to be 1/n^2. I am sure that 1/n^2 converges but does it work as a bound on each term? And then what should I do with these two (the series and the sequence of Mk.)?

2. Feb 24, 2009

### Office_Shredder

Staff Emeritus
Looks like it does

$$| \frac{1}{n^2} cos(nx) | = \frac{1}{n^2} |cos(nx)|$$

and |cos(nx)| is always less than or equal to 1.

3. Feb 25, 2009

### letmeknow

That wasn't too bad. But I have a more specific question. Does the M test say that the domain of convergence is R? Since are sequence of functions is defined on R?

4. Feb 25, 2009

### Office_Shredder

Staff Emeritus
Yes. The M test says the series converges uniformly on R, so in particular converges on R.