Results 1 to 6 of 6

Math Help - Real-valued function

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    14

    Real-valued function

    Hi ! Do you know how to solve this problem:

    Find a real-valued function on R possessing derivatives of all orders whose Taylor series at a certain point converges to the function only at that point.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Milus View Post
    Hi ! Do you know how to solve this problem:

    Find a real-valued function on R possessing derivatives of all orders whose Taylor series at a certain point converges to the function only at that point.
    Consider x^{x^x} at x=0
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    14
    Thank,but still I do not know what means that up side down L inside the log parentasice.
    Also, do you have any idea how to prove it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Milus View Post
    Thank,but still I do not know what means that up side down L inside the log parentasice.
    Also, do you have any idea how to prove it.
    That is my signature, that is not part of the post. And for the proof try finding the general form for the n-th derivative of x.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2008
    Posts
    14
    Can you help me a little more?
    I am still strugling with general form of the derivative
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by Milus View Post
    Hi ! Do you know how to solve this problem:

    Find a real-valued function on R possessing derivatives of all orders whose Taylor series at a certain point converges to the function only at that point.
    I didn't know MathStud28's example. The example that I would have called the "classical" one is: f(x)=e^{-\frac{1}{|x|}} for x\neq 0 and f(0)=0. I can't remember which famous mathematician introduced it, perhaps Cauchy, anyway it has a long history. And it seems simpler to deal with.

    For x>0, f(x)=e^{-\frac{1}{x}}. First notice that f(x)\to_{x\to 0^+} 0. Notice as well \frac{e^{-\frac{1}{x}}}{x^n}\to_{x\to 0^+}0 for any integer n. To convince yourself of this, substitute u=\frac{1}{x}, so that the limit is \lim_{u\to\infty} u^n e^{-u}.
    Now, the n-th derivative of f is (easily) seen to be a rational function of x (a quotient of two polynomials) times e^{-\frac{1}{x}}. This is proved by induction (suppose there is a rational function R such that the n-th derivative is, for x>0, R(x)e^{-\frac{1}{n}}, and prove that the next derivative is again of the same kind).
    Because of the previously mentioned limit, you obtain that the n-th derivative converges to 0 as x tends to 0 from the right. By symmetry, the same holds from the left.
    Finally, there is a theorem (that you probably know if you're asked this problem) telling you that, as a consequence, f is indefinitely differentiable at 0 and the derivatives at 0 are the limits of the derivatives at x when x tends to 0. That is to say, all derivatives at 0 exist and are 0. Yet, the function is only zero at zero...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. an extended real-valued function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 21st 2010, 10:28 PM
  2. continuity for a real-valued function
    Posted in the Differential Geometry Forum
    Replies: 15
    Last Post: February 23rd 2010, 07:27 AM
  3. Real Valued Function
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 22nd 2008, 04:01 AM
  4. Real valued function on [0,1]
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 15th 2008, 10:34 AM
  5. real valued function
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 26th 2008, 05:05 AM

Search Tags


/mathhelpforum @mathhelpforum