Attempt at a rigorous (dis)proof of uniform convergence

[end3r7]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

$$|\frac{1}{{n^{x}\ln (x)}}| > \frac{1}{|{n||\ln (x)|}} > |\frac{ne}{n}| = e$$

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) ??