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Math Help - Series - Converge or diverge

  1. #1
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    Series - Converge or diverge

    Hello,

    How can I show if these series are convergent or divergent?

    k = 1∞ ((k!)^(1/k))/(k)


    For the following series:

    k = 1∞ (e^(3ik))/(k^(3/2))

    I have started a solution here:http://img819.imageshack.us/img819/6864/solution.jpg
    and was wondering if it is correct. Why is (e^(3ik))=1 when k->∞


    Appriciate any guidance and a detail explanation.

    Thank you
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  2. #2
    Member mohammadfawaz's Avatar
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    |e^{3ik}|= 1 because it's magnitude is 1.
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  3. #3
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    Quote Originally Posted by 4Math View Post
    Hello,

    How can I show if these series are convergent or divergent?

    k = 1∞ ((k!)^(1/k))/(k)


    For the following series:

    k = 1∞ (e^(3ik))/(k^(3/2))

    I have started a solution here:http://img819.imageshack.us/img819/6864/solution.jpg
    and was wondering if it is correct. Why is (e^(3ik))=1 when k->∞


    Appriciate any guidance and a detail explanation.

    Thank you

     \sqrt[k]{k!} > 1 so

     \frac{\sqrt[k]{k!} }{k} > \frac{1}{k} > 0
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  4. #4
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    Quote Originally Posted by simplependulum View Post
     \sqrt[k]{k!} > 1 so

     \frac{\sqrt[k]{k!} }{k} > \frac{1}{k} > 0
    Is this mean that the seies convergent or divergent??
    Last edited by 4Math; August 6th 2010 at 11:58 AM.
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  5. #5
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    Does \displaystyle \sum\limits_k {\frac{1}{k}} converge or diverge?
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