Attempt at a rigorous (dis)proof of uniform convergence

In summary, the conversation discusses testing a series for uniform convergence on the interval [0,1]. The attempt at a solution includes showing that the series is not continuous and discussing the validity of using x values close to 1. However, it is concluded that the fact that the series is not even defined at 0 and 1 is enough to determine that it does not converge on [0,1]. The question is also posed whether the interval should be (0,1) instead of [0,1].
  • #1
end3r7
171
0

Homework Statement


1) Test the following series for Uniform Convergence on [0,1]
[tex]
\sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n^{x}\ln (x)}}}
[/tex]


Homework Equations





The Attempt at a Solution



Obviously, it's not uniformly convergent since f(n,1) = [tex]
\sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n\ln (1)}}}
[/tex] is not continuous. I'm looking for a more rigorous proof though.

I'm wondering if thsi reasoning is correct:

What I did was show that I can get x arbitrarily close to 1, and since log(x) is continuous, I can get arbitrarily close 1. Basically, I showed that for any 'n', I can make [tex]
log(x) < \frac{1}{ne}
[/tex]
so for x sufficiently close to 1

[tex]
|\frac{1}{{n^{x}\ln (x)}}| > \frac{1}{|{n||\ln (x)|}} > |\frac{ne}{n}| = e
[/tex]

where first inequality holds because x is between 0 and 1.

Therefore the terms of the series do not go to zero uniformly on its domain, so it can't converge.

Is that a valid argument?
 
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  • #2
The fact that the series is not even defined at 0 and 1 is enough to conlude that it does not converge (uniformly or not) on [0,1].

Are you sure the question asks you to discusss uniform converge on [0,1] and not (0,1) ??
 

Related to Attempt at a rigorous (dis)proof of uniform convergence

1. What is uniform convergence?

Uniform convergence is a type of convergence in mathematics where a sequence of functions converges to a single function in such a way that the convergence is independent of the point at which the function is evaluated. This means that the rate of convergence is the same at all points.

2. How is uniform convergence different from pointwise convergence?

In pointwise convergence, the convergence of a sequence of functions depends on the point at which the function is evaluated. This means that the rate of convergence may vary at different points. In uniform convergence, the rate of convergence is the same at all points, making it a stronger form of convergence.

3. What is the importance of uniform convergence?

Uniform convergence is important in mathematics because it allows us to extend the properties of a sequence of functions to the limit function. This means that we can use the properties of the individual functions in the sequence to infer properties of the limit function.

4. How is uniform convergence related to continuity?

In mathematics, a function is continuous if small changes in the input result in small changes in the output. Uniform convergence is closely related to continuity because it ensures that the limit function is continuous if the individual functions in the sequence are continuous.

5. What are some common techniques used to prove uniform convergence?

Some common techniques used to prove uniform convergence include the Cauchy criterion, the Weierstrass M-test, and the uniform limit theorem. These techniques involve showing that the rate of convergence of the sequence of functions is bounded and independent of the point at which the function is evaluated.

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