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Math Help - Finding limit as n approaches infinity

  1. #1
    xyz
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    Finding limit as n approaches infinity



    I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by xyz View Post


    I am not able to prove it by the hint shown. I perfectly understand the question and it is obvious that the antiderivative approaches 0 as n approaches infinity. But the hint about riemann integral is confusing me.
    The idea is that you want to prove that if f_n=\frac{1}{(1+x^2)^n}, then f_n\to0 uniformly on [1,2]. Then you know that you can interchange the limits, i.e:

    \lim_{n\to\infty}\int_1^2(1+x^2)^{-n}\,dx=\int_1^2\left(\lim_{n\to\infty}(1+x^2)^{-n}\right)\,dx=\int_1^2 0\,dx=0

    So all you need to prove is uniform convergence, and then the above equality will hold. (Hint: Use Dini's Theorem.)
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