# Valid application of Weierstrass Test?

1. Mar 4, 2008

### end3r7

It would seem so

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

If
$$\sum\limits_{n = 1}^{\inf } {|{f(n,x)}|}$$ is uniformly convergent on [a,b], then is $$\sum\limits_{n = 1}^{\inf } {{f(n,x)}}$$ uniformly convergent.

2. Relevant equations

3. The attempt at a solution

I said yes. And just applied the Weierstrass Test with |f(n,x)| <= |f(n,x)| (a basic comparison test)

Should still be valid right? Since the absolutely value is Uniformly Cauchy

2. Mar 4, 2008

### quasar987

Just because a series converge uniformly does not mean it satisfy Weierstrass' M test

3. Mar 5, 2008

### end3r7

Well, forget the Weierstrass test then...
If one is uniformly Cauchy, wouldn't that make the toher essentially uniformly Cauchy as well?

4. Mar 5, 2008

### quasar987

This sounds right! (triangle inequality)

5. Dec 12, 2011

### Xiola168

How does uniformly Cauchy apply to absolute uniform convergence