- #1

end3r7

- 171

- 0

Now, I am almost 100% sure this is not uniformly convergence at x=1, but since it's a stinking series, it's hard to figure out it's summation

## Homework Statement

data[/b]

1) Test the following series for Uniform Convergence on [0,1]

[tex]

(1-x)\sum\limits_{n = 1}^{\inf } {\frac{x^n}{1+x^n}}

[/tex]

## The Attempt at a Solution

I honestly have not a clue where to start. I want to try the following, and I don't know if it's true.

[tex]

\frac{1}{2} < \frac{1}{1+x^n}

[/tex]

for x in [0,1)

So [tex]

(1-x)\sum\limits_{n = 1}^{\inf } {\frac{x^n}{2}} <

(1-x)\sum\limits_{n = 1}^{\inf } {\frac{x^n}{1+x^n}}

[/tex]

and for x in [0,1) [tex]

(1-x)\sum\limits_{n = 1}^{\inf } {\frac{x^n}{2}} = \frac{1}{2}

[/tex]

and we know that and for x = 0 [tex]

(1-x)\sum\limits_{n = 1}^{\inf } {\frac{x^n}{1+x^n}} = 0

[/tex]

Thus the limit function is not continuous, so it can't be uniformly convergent.

Sound proof or bad proof?