Results 1 to 5 of 5

Math Help - show uniform convergence of sequence of functions

  1. #1
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53

    show uniform convergence of sequence of functions

    The exercise is to show that the sequence x^2 e^{-nx} converges uniformly on [0, \infty)

    I know that the sequence converges to 0 because x^2 e^{-nx} = x^2( e^{-x})^n and e^{-x} is always less than 1.

    It makes sense that it converges uniformly since e^{-nx} will go to 0 as n \rightarrow \infty, not depending on what x is, but I don't know how to show this using the definition of uniform convergence. Help is appreciated. Thank you!
    Last edited by mremwo; April 18th 2011 at 08:25 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    Spoken roughly, once it get's close, it never wanders off again. You must demonstrate, since you seem confident this converges to f(x) = 0 on the interval, that for some value of n, it is very close to zero on the entire interval AND that it never gets any worse than that.

    For your sequence, the maximum deviation is always at x = 2/n. Does that do anything for us? Maximum deviation is 4/[(e^2)*(n^2)].
    Last edited by TKHunny; April 18th 2011 at 10:45 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53
    So the sequence is farthest from 0 at x=2/n, or the maximum of each function in the sequence is at x=2/n? So if the maximum deviation of the functions in the sequence is 4/[(e^2)*(n^2)] for each n, I need to make sure my natural number K makes it so that if n>=k then |sequence-0|<= 4/[(e^2)*(n^2)] < E ??
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by mremwo View Post
    So the sequence is farthest from 0 at x=2/n, or the maximum of each function in the sequence is at x=2/n? So if the maximum deviation of the functions in the sequence is 4/[(e^2)*(n^2)] for each n, I need to make sure my natural number K makes it so that if n>=k then |sequence-0|<= 4/[(e^2)*(n^2)] < E ??
    Merely note that x^2e^{-nx}= x^2/e^{nx}=x^2/{1+nx+n^2x^2/2+...}<= x^2/{1+n^2x^2} and I think it's easy to see the result from there. Namely, by Dini's theorem it's evident that since this converges pointwise to 0 on [0,1] that it converges uniformly there and on (1,∞) one has that x^2/e^{nx}<=x^2/{1+nx^2}<= 1/n
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by mremwo View Post
    The exercise is to show that the sequence x^2 e^{-nx} converges uniformly on [0, \infty)

    I know that the sequence converges to 0 because x^2 e^{-nx} = x^2( e^{-x})^n and e^{-x} is always less than 1.

    It makes sense that it converges uniformly since e^{-nx} will go to 0 as n \rightarrow \infty, not depending on what x is, but I don't know how to show this using the definition of uniform convergence. Help is appreciated. Thank you!
    Each is greater or equal to 0 in the whole interval and has a maximum at
    where is . If You take an then in the whole interval it will be ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 31st 2010, 08:09 PM
  2. Uniform convergence of a function sequence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 3rd 2010, 09:23 AM
  3. Uniform Convergence of the Sequence of Compositions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 5th 2010, 09:33 PM
  4. Uniform convergence of a sequence of functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 17th 2010, 01:49 PM
  5. Show sequence convergence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 25th 2009, 09:27 PM

Search Tags


/mathhelpforum @mathhelpforum