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Math Help - Limit of the real sequence nk^n

  1. #1
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    Limit of the real sequence nk^n

    I am trying to prove the following limit in formal terms.

    lim n -> inf (n x^n) = 0, where |x| < 1


    I dont want to use the L'hopitals rule or any other method that involves differentiation. I am trying to use the following sequence definition.

    x --> x0 =
    (0 < e ==> there exists and N. such that for all n. n >= N ==>
    abs (x n - x0) < e

    but have got stuck in the middle.

    Any hints regarding this would be appreciated.
    Thanks,
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  2. #2
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    Quote Originally Posted by led5v View Post
    I am trying to prove the following limit in formal terms.

    lim n -> inf (n x^n) = 0, where |x| < 1
    You want to compute,
    lim inf n*x^n

    Instead if we can show,
    lim n*x^n =0
    Then the lim inf is the same.

    Note,
    lim n*x^n =0

    Hint: Use the ratio test (for x!=0).
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