- #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?