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Math Help - Convergence problem

  1. #1
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    Convergence problem

    If anyone could explain how the following problem is done, it would be greatly appreciated.

    Show that the following series either converges or diverges by using a suitable test:

    sigma starting at j=2 and going to infinity of:
    (-1)^j * ln(j)/j
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  2. #2
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    Quote Originally Posted by clockingly View Post
    If anyone could explain how the following problem is done, it would be greatly appreciated.

    Show that the following series either converges or diverges by using a suitable test:

    sigma starting at j=2 and going to infinity of:
    (-1)^j * ln(j)/j
    Use Leibniz Alternating Series Test.
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  3. #3
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    Quote Originally Posted by clockingly View Post
    If anyone could explain how the following problem is done, it would be greatly appreciated.

    Show that the following series either converges or diverges by using a suitable test:

    sigma starting at j=2 and going to infinity of:
    (-1)^j * ln(j)/j
    An alternating series converges if eventualy the absolute values of the terms
    are monotonicaly decreasing with limit 0.

    Consider:

    f(x) = ln(x)/x

    f'(x) = 1/x^2 - ln(x)/x^2

    and for x>3 ln(x)>1, so f'(x)<0 for x>3 and so f(x) is decreasing for x>3.
    So the absolute values of the terms of your series are decreasing for j>3.

    Now it remains to show that lin_{j-> infty} ln(j)/j = 0, which can be done
    using L'Hopital's rule on f(x)

    RonL

    RonL
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