# Showing a function converges uniformly

1. Nov 16, 2013

### trap101

let fn(x) = n/(1+n2x2) - (n-1)/(1+(n-1)2x2) in the interval 0<x< L

I am trying to show that this series converges uniformly.

I have solved that the sum of the series from n=1 to n = N is:

N/(1+N2x)

now by definition a series converges uniformly if:

max (a≤x≤b) |f(x) - Sn(x)| ---> 0 as N-->∞

my issue is that in the solution example they provided in the textbook they said the series does not uniformly converge because:

max(0,L) 1/(1+N2x2) = N

how did they get 1/(1+N2x2) from the definition of uniform convergence? What is it that I am not interpreting right to get a solution?

2. Nov 16, 2013

### LCKurtz

On what interval? Are you sure you stated the interval correctly? Was it [a,L) for any a between 0 and L and you restated it (0,L)?

That isn't true on (0,L). It's true on [0,L]. On (0,L) is is a sup, not a max.

I don't know where those last two lines came from. And I would like to see the exact statement of the problem. I don't believe it is true as you have stated it.

3. Nov 16, 2013

### trap101

You were right in terms of the interval supposed to be closed on [0,L] as well as the exact definition of uniform convergence I just didn't communicate it correctly. If I polished up on my latex it would be easier for me to restate. I attached a jpeg of the question and statement that is confusing me so you can see the exact question.

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4. Nov 16, 2013

### LCKurtz

That last equation$$\max_{(0,L)}\frac 1 {1+N^2x^2}=N$$obviously has typos and should read$$\max_{[0,L]}\frac N {1+N^2x^2}=N$$

5. Nov 16, 2013

### trap101

I'm still having issues with how they got N, because are we not evaluating the difference between f(x) and the sum of the series? I don't see how that difference translates into just N.

6. Nov 16, 2013

### LCKurtz

OK, maybe they meant to write$$\sup_{(0,L)}\frac N {1+N^2x^2}=N$$The point is that although you have pointwise convergence to $0$ it is not uniform because the larger N is the bigger the sum is near zero. You can't uniformly bound the sum near 0 no matter how large N is.

7. Nov 16, 2013

### trap101

Ahhhh, now I see. Thanks.