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Math Help - Domain of Convergence

  1. #1
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    Domain of Convergence

    I am given the problem:

    Show that r^{ln(n)} = n^{ln(r)} and determine the values of r (r > 0) for which the series \sum_{n=1}^{\infty} r^{ln(n)} converges.

    I guess my initial problem is to show that the two are equal to one another as such. However, I thought that the minimum value of r that would converge would be 0 and the max would be 1.

    If someone could just walk through the initial part of:

    r^{ln(n)} = n^{ln(r)}

    I would be very greatful. Thanks guys.
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  2. #2
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    Quote Originally Posted by Spudwad View Post
    I am given the problem:

    Show that r^{ln(n)} = n^{ln(r)} and determine the values of r (r > 0) for which the series \sum_{n=1}^{\infty} r^{ln(n)} converges.

    I guess my initial problem is to show that the two are equal to one another as such. However, I thought that the minimum value of r that would converge would be 0 and the max would be 1.

    If someone could just walk through the initial part of:

    r^{ln(n)} = n^{ln(r)}

    I would be very greatful. Thanks guys.

    It follows at once from the very definition of logarithms that a^b=e^{b\ln a}\,,\,\,a>0 . Now apply this to both sides of the given equality and verify it is really an equality.

    Tonio
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