1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Applying the Mean Value Theorem to sequences of function

  1. Jan 18, 2012 #1
    As usual, I typed up the problem and my attempt in LaTeX:

    MVT.png

    Maybe I'm not applying the MVT correctly, but my result does not seem to help me solve the problem in anyway. What are your thoughts?
     
  2. jcsd
  3. Jan 18, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Maybe you shouldn't apply the mean value theorem on [a,b] but rather on [x,x0]??
     
  4. Jan 19, 2012 #3
    Thanks, that would help. Ok I tried your suggestion and here is the result. The problem is now making use of the [itex]f'_n(c) - f'_m(c)[/itex] since they do not appear in the triangle inequality. And also, I am not using the assumption that [itex]f'_n[/itex] converges uniformly.

    MVT-1.png
     
  5. Jan 19, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    In your triangle inequality, you have a term

    [tex]|(f_n(x)-f_m(x))-(f_n(x_0)-f_m(x_0))|[/tex]

    Isn't this the same as the numerator of what you got after applying the mean value theorem??
     
  6. Jan 20, 2012 #5
    Well, no, it's slightly different:

    [tex]|f_n(x) + f_m(x) - f_n(x_0) - f_m(x_0)|[/tex]

    We need it to be [tex]f_n(x) - f_m(x)[/tex] instead of +

    Oh btw, I noticed in your profile that it says you are 15. Is that true? Because that's incredible that you know so much math at that age, really astonishing.
     
  7. Jan 20, 2012 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Oh, how did you get that + in the mean value theorem? That isn't correct. The mean value theorem says that

    [tex](f_n-f_m)^\prime(c)(x-x_0)=(f_n-f_m)(x)-(f_n-f_m)(x_0)[/tex]

    You seemed to have applied the mean-value theorem to [itex]f_n+f_m[/itex]...
     
  8. Jan 22, 2012 #7
    Micromass, I believe I have found the solution! Here is my completed proof, what do you think?


    MVT-2.png
     
  9. Jan 22, 2012 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Hmm, you know that that c is depend on n and m right?? You act like there is only one c, but there are multiple c's.

    And to be sure you really get it: where exactly do you need uniform convergence of [itex](f^\prime_n)_n[/itex]??
     
  10. Jan 22, 2012 #9
    Oh, well it was my understanding that the c was only dependent on the [tex][x, x_0][/tex]. I guess I didn't understand that c could be different for different m's and n's.

    We need uniform convergence of [itex](f^\prime_n)_n[/itex] because we have to use the Cauchy Criterion for Uniform Convergence.

    Well, since my proof is incorrect then, could you point me in the right direction so that I can try to complete this?
     
  11. Jan 22, 2012 #10

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    The mean value theorem on a function actually implies:

    [tex]|g(a)-g(b)|\leq |a-b| \sup_{c\in [a,b]}|g^\prime(c)|[/tex]

    I suggest you use this. It will become apparent where the uniform convergence is used.
     
  12. Jan 22, 2012 #11
    I guess I'm not quite sure I understand. My book has no mention of this variation to the MVT. Are you certain that it is necessary to solve this problem?
     
  13. Jan 22, 2012 #12

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    It's quite easy to prove from the MVT.

    I think it's the easiest and cleanest way...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Applying the Mean Value Theorem to sequences of function
Loading...