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Math Help - Sequence of Functions Convergence

  1. #1
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    Sequence of Functions Convergence

    I'm not too sure how to do this, so any help would be great:

    fn(x) = n/(1+nx^2) on (0, infinity)

    a) find the pointwise limit
    b)prove (or disprove) if the sequence converges uniformly.

    I don't really know how to find the pointwise limit because I can't factor the denominator, and I can't separate the numerator into two fractions (like the examples in the book).

    Also, I have another one that is a bit different:

    fn(x) = sin(nx)/1+nx on [1,2].
    How do I figure out the pointwise limit on that domain?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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     \frac{n}{1+nx^2} = \frac{n}{n} \cdot \frac{1}{\frac{1}{n}+x^2} = \frac{1}{\frac{1}{n}+x^2}.

    Hence  \frac{n}{1+nx^2} \longrightarrow \frac{1}{x^2} pointwise.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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     -\frac{1}{1+nx} \leq \frac{\sin(nx)}{1+nx} \leq \frac{1}{1+nx} .

    Now apply the squeeze theorem.
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