Showing a function converges uniformly

[trap101]

let f_n(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) - S_n(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?

 

[LCKurtz]

let f_n(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.

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)?

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) - S_n(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

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

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?

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.

 

[trap101]

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.

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.

 

[LCKurtz]

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

 

[trap101]

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

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.

 

[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.

 

[trap101]

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.

Ahhhh, now I see. Thanks.