Results 1 to 13 of 13
Like Tree6Thanks
  • 1 Post By girdav
  • 1 Post By girdav
  • 1 Post By girdav
  • 1 Post By girdav
  • 1 Post By girdav
  • 1 Post By girdav

Math Help - function sequence convergence

  1. #1
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    function sequence convergence

    State where

    f_n(x)=x^{2n}-x^{n}, x \in \left[ -1, 1\right]

    is pointwisely convergent and find a limit function.

    then assess if f_n(x)=x^{2n}-x^{n}, x \in \left[ -1, 1\right] is uniformly convergent for x \in \left[ 0,  \frac{1}{2} \right]

    please help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    Hint for the pointwise limit: deal with the cases x=1, x=-1, x\in (-1,1).
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    Quote Originally Posted by girdav View Post
    Hint for the pointwise limit: deal with the cases x=1, x=-1, x\in (-1,1).
    so in case of x=-1 there is no limit. when x \in (-1,1] limit is equal to 0

    yes?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    That's correct.
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    So it isnt uniformly convergent when x \in [-1,1], but it may be when x \in [0,1/2] , yes?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    Yes, and that's what you have to determine.
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    sup|x^{2n} - x^{n}| \rightarrow 0 , when x \in [0, 1/2] and n \rightarrow \infty

    So in that interval the function sequence is uniformly convergent. Is that ok and enough writing for an exam?
    Last edited by fqqs; May 26th 2012 at 11:33 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    Maybe you could justify that \sup_{0\leq x\leq 1/2}|x^{2n}-x^n|\to 0, for example using an upper bound.
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    Quote Originally Posted by girdav View Post
    Maybe you could justify that \sup_{0\leq x\leq 1/2}|x^{2n}-x^n|\to 0, for example using an upper bound.
    could you explain?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    I just meant that \sup_{0\leq x\leq 1/2}|x^{2n}-x^n|\leq \frac 1{2^n} in order to justify uniform convergence (just saying it is not enough).
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    Maybe Im stupid but I just cant see why it is so...
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    678
    Thanks
    32

    Re: function sequence convergence

    For 0\leq x\leq \frac 12 and n integer |x^{2n}-x^n|=|x|^n|x^n-1|\leq \frac 1{2^n}(1-x^n)\leq \frac 1{2^n}.
    Thanks from fqqs
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Newbie
    Joined
    May 2012
    From
    poland
    Posts
    21

    Re: function sequence convergence

    aaa of course! thank you very much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniform convergence of a function sequence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 3rd 2010, 08:23 AM
  2. Replies: 7
    Last Post: October 12th 2009, 10:10 AM
  3. Replies: 6
    Last Post: October 1st 2009, 09:10 AM
  4. Replies: 6
    Last Post: October 24th 2008, 01:45 PM
  5. Replies: 3
    Last Post: October 6th 2007, 02:01 PM

Search Tags


/mathhelpforum @mathhelpforum