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Math Help - limit sequence harmonic

  1. #1
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    limit sequence harmonic

    Hi,

    How do I show that the limit of (ln n)/(n^p) as n -> infinity, where p>0, converges to 0? I know that the denominator is the harmonic sequence (1/n^p) but I am not sure how to begin.

    Thanks.
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  2. #2
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    Re: limit sequence harmonic

    How about L'Hopital's rule?

    - Hollywood
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  3. #3
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    Re: limit sequence harmonic

    Ah yes I get it now. Will the same apply for:

    (n^p)/(a^n) where a>1 and p>0 as n -> infinity and

    (a^n)/(n!) where a>1 as n -> infinity?
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  4. #4
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    Re: limit sequence harmonic

    I think you can use l'Hopital's rule for \frac{n^p}{a^n}, but not \frac{a^n}{n!}. For the latter, my first thought would be mathematical induction.

    - Hollywood
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  5. #5
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    Re: limit sequence harmonic

    a^n and n! both have n factors but for, say, n= 4a, 3/4 of the factors in n! are larger than those in [tex]a^n[/itex]
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  6. #6
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    Re: limit sequence harmonic

    Quote Originally Posted by hollywood View Post
    How about L'Hopital's rule?

    - Hollywood
    You cannot use L'hospital's rule because the numerator and denominator are not equal to zero in the limit.
    Also the function is not differentiable because it is only defined for n being natural number.

    Is p a natural number? because it doesnt converge for 1>p>0
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