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Math Help - Converge and diverge of an improper integral

  1. #1
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    Converge and diverge of an improper integral

    For which values of k , where k is a Real number does the improper integral:

    integral( (ln(x)^k) / (x) ) from 1 to infinity

    converge? For which values of k does it diverge? Justify your answer.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Chris23 View Post
    For which values of k , where k is a Real number does the improper integral:

    integral( (ln(x)^k) / (x) ) from 1 to infinity

    converge? For which values of k does it diverge? Justify your answer.
    Consider that \forall{k}\in[1,\infty)~\frac{\ln^k(x)}{x}\geqslant\frac{1}{x} which diverges. Now I want you to try out the other answers, the most important other number is when k=-1 Remember that you can integrate this as it is/.
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    RE: Converge or diverge of an improper integral

    In the exercise I meant that k is a real number, can take any value from -infinity to +infinity. The improper integral has lower limit 1 and upper limit +infinity.
    Do you know if there is any specific range in which the improper integral diverges and any other range that the improper definite integral converges?
    And who I can find this? By taking limits?

    Thanks a lot!
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Chris23 View Post
    In the exercise I meant that k is a real number, can take any value from -infinity to +infinity. The improper integral has lower limit 1 and upper limit +infinity.
    Do you know if there is any specific range in which the improper integral diverges and any other range that the improper definite integral converges?
    And who I can find this? By taking limits?

    Thanks a lot!
    What I said was in response to it being a definite integral, I did not explicity state this but what I meant to say was that

    \int_1^{\infty}\frac{\ln^k(x)}{x}dx\geqslant\int_1  ^{\infty}\frac{dx}{x}=\infty~\forall{k\geqslant{0}  }
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