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Math Help - Uniform Convergence

  1. #1
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    Uniform Convergence

    Consider the sequence of functions f_n (x)= x / (1+nx^2)

    let f = lim f_n. Compute f'_n and find all values of x such that f'(x)=lim f'_n (x)
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  2. #2
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    Re: Uniform Convergence

    Can you state 'f' explicitly? Is it constant, linear, what?

    Can you find f'_n, generally?
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    Re: Uniform Convergence

    f_'n (x)=(1-nx^2)/(1+nx^2 )^2 and f=0
    Last edited by veronicak5678; November 16th 2011 at 06:58 PM.
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    Re: Uniform Convergence

    Very good. You're almost there. Where is the derivative zero?
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    Re: Uniform Convergence

    The derivative is zero when nx^2 =1, or when x=(1/n)^1/2

    But isn't lim f' =0 for all x?
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    Re: Uniform Convergence

    Let's try to remember which variable is moving. Are we talking about x approaching zero or n increasing without bound?

    AT x = 0, what is the limit of f'_n as n increases without bound?
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    Re: Uniform Convergence

    At x=0, f'_n =1
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  8. #8
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    Re: Uniform Convergence

    Right, and that's not zero (0), is it?

    I believe you were done with this problem when you said this: "x=(1/n)^(1/2)"
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  9. #9
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    Re: Uniform Convergence

    I guess I thought there was more to it. Thanks for your help.
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