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Showing a function converges uniformly

  1. Nov 16, 2013 #1
    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. jcsd
  3. Nov 16, 2013 #2

    LCKurtz

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    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.
     
  4. Nov 16, 2013 #3



    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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  5. Nov 16, 2013 #4

    LCKurtz

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    That last equation$$
    \max_{(0,L)}\frac 1 {1+N^2x^2}=N$$obviously has typos and should read$$
    \max_{[0,L]}\frac N {1+N^2x^2}=N$$
     
  6. Nov 16, 2013 #5


    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.
     
  7. Nov 16, 2013 #6

    LCKurtz

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    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.
     
  8. Nov 16, 2013 #7

    Ahhhh, now I see. Thanks.
     
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