Attempt at a rigorous (dis)proof of uniform convergence

[end3r7]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Obviously, it's not uniformly convergent since f(n,1) = <br /> \sum\limits_{n = 1}^{\inf } {\frac{{( - 1)^n }}{{n\ln (1)}}} <br /> 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 <br /> log(x) &lt; \frac{1}{ne} <br />
so for x sufficiently close to 1

<br /> |\frac{1}{{n^{x}\ln (x)}}| &gt; \frac{1}{|{n||\ln (x)|}} &gt; |\frac{ne}{n}| = e<br />

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?

 

[quasar987]

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) ??