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Math Help - Series of functions & uniform convergence

  1. #1
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    Arrow Series of functions & uniform convergence



    OK, so the answer I got for part a is:
    S(x) = x^2 / [1- 1/(1+ x^2)] = 1 + x^2 if x≠0
    S(x) = 0 if x=0


    Can someone help me with the part b, please? In general, I am having a lot of headaches on problems about uniform convergence. I know the precise definition of it, and I re-read the definition many times, but I have no idea how to actually APPLY the definition to solve actual problems.
    Should we use the Weierstrass M-test here? Is this the only way to prove that a SERIES of functions is uniformly convergent?

    Thanks for any help!!


    [also under discussion in math link forum]
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post


    OK, so the answer I got for part a is:
    S(x) = x^2 / [1- 1/(1+ x^2)] = 1 + x^2 if x≠0
    S(x) = 0 if x=0


    Can someone help me with the part b, please? In general, I am having a lot of headaches on problems about uniform convergence. I know the precise definition of it, and I re-read the definition many times, but I have no idea how to actually APPLY the definition to solve actual problems.
    Should we use the Weierstrass M-test here? Is this the only way to prove that a SERIES of functions is uniformly convergent?

    Thanks for any help!!


    [also under discussion in math link forum]
    Hint:
    Spoiler:
    f_n(x)\leqslant\frac{\left(1-\frac{1}{n}\right)^n}{n}.
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  3. #3
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    Sorry, I don't understand...

    How did you get that inequality and how is that going to help?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post
    How did you get that inequality
    Use calculus (max and mins)

    and how is that going to help?
    Weierstrass M-test.
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  5. #5
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    Sorry, I still don't get it. That bound is not too obvious/natural to me.

    For part b, what interval [a,b] should we work with? In what interval does it converge uniformly?

    I hope someone can help me out! Thank you!
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kingwinner View Post
    Sorry, I still don't get it. That bound is not too obvious/natural to me.
    I used simple calculus. In other words, I calculated f'_n(x) set it equal to zero..blah blah blah.

    For part b, what interval [a,b] should we work with? In what interval does it converge uniformly?
    What are you thinking?

    I hope someone can help me out! Thank you!
    I'm saying this in passing, but you say stuff like this quite often in all your threads and it makes it sound as though you are disregarding those who have helped you. It could be offensive to some! just saying
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