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Math Help - Uniform convergence

  1. #1
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    Uniform convergence

    Hello, I need to investigate Uniform convergence for these two using Weierstrass M-test, 10x in advance.
    Assaf.
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  2. #2
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    For the first problem note x^n + x^{-n} \leq 2.
    Thus, |f_n(x)| \leq \frac{2n^2}{\sqrt{n!}} and \sum_{n=0}^{\infty} \frac{2n^2}{\sqrt{n!}} < \infty.

    I did not do the second problem but I would imagine you need the fact \log \left( 1 + \frac{x^2}{n\log^2 n} \right) \leq \log \left( 1+\frac{a^2}{n\log^2 n} \right).
    Now let us hope that \sum_{n=2}^{\infty} \log \left( 1+\frac{a^2}{n\log^2 n} \right) < \infty.
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  3. #3
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    it is

    is is < Inf, I used my calculator to see, the question is how I show it...
    10x.
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    Quote Originally Posted by asi123 View Post
    is is < Inf, I used my calculator to see, the question is how I show it...
    10x.
    It is! And there is an easy way to show it.

    You need the fact that \log (1+x) \leq x for x\geq 0.
    Thus,
    0\leq \log \left( 1 + \frac{a^2}{n^2\log n} \right) \leq \frac{a^2}{n^2\log n}
    And of course, by applying Betrand's test
    \sum_{n=2}^{\infty}\frac{a^2}{n^2\log n} < \infty
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    For the first problem note x^n + x^{-n} \leq 2.
    x^n + x^{-n} \leq 2
    How is it true generally?
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  6. #6
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    Quote Originally Posted by Isomorphism View Post
    x^n + x^{-n} \leq 2
    How is it true generally?
    It should be x^n + x^{-n}\geq 2. But that does not solve the problem then.
    In that case we should maybe consider x^n+x^{-n} \leq 2^n+2^{-n}.
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