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Math Help - Find all positive values of b for which the following series is convergent.

  1. #1
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    Find all positive values of b for which the following series is convergent.

    hi

    sigma from to infinity for b^(ln n)

    hmmm i tried to write :
    b^(ln n) = [ ( b^(ln n) )^1/n ] ^ n
    and deal with it as an goemetric series .. but it failed ;p

    help :/
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  2. #2
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    Quote Originally Posted by TWiX View Post
    hi

    sigma from to infinity for b^(ln n)

    hmmm i tried to write :
    b^(ln n) = [ ( b^(ln n) )^1/n ] ^ n
    and deal with it as an goemetric series .. but it failed ;p

    help :/
     0 \leq b < e^{-1}
    Last edited by mr fantastic; September 18th 2009 at 08:45 AM. Reason: Restored original reply.
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  3. #3
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    Can you give me its idea ?
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    Quote Originally Posted by TWiX View Post
    Can you give me its idea ?
    Apply the ratio test and force the limit to be less than 1 to get the radius of convergence. Then test the endpoints to get the interval of convergence.
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    Is this solution work here ?

    b^(ln n) = [ e^(ln b) ]^(ln n) = [ e^(ln n) ]^(ln b) = n^(ln b) = 1/[ n^(ln(1/b)) ]
    which is a p-series.

    ln(1/b) > 1 ----> 1/b > e ----> 0 < b < 1/e
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    Apply the ratio test and force the limit to be less than 1 to get the radius of convergence. Then test the endpoints to get the interval of convergence.
    Edit:
    your solution is wrong
    the limit is 1
    test failed
    i noticed this now.
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  7. #7
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    Quote Originally Posted by TWiX View Post
    Is this solution work here ?

    b^(ln n) = [ e^(ln b) ]^(ln n) = [ e^(ln n) ]^(ln b) = n^(ln b) = 1/[ n^(ln(1/b)) ]
    which is a p-series.

    ln(1/b) > 1 ----> 1/b > e ----> 0 < b < 1/e
    Yes, that's a very good solution
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