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Problem with Series.

  1. Nov 18, 2007 #1
    1. The problem statement, all variables and given/known data

    Determine whether the series converges or diverges.


    [tex]\sum_{n=1}^{\infty}\frac{1}{n^{1+\frac{1}{n}}}[/tex]

    3. The attempt at a solution

    [tex]\sum_{n=1}^{\infty}\frac{1}{nn^{\frac{1}{n}}}=\sum_{n=1}^{\infty}\frac{1}{ne^{\frac{1}{n}\ln n}}[/tex]

    [tex]\lim_{n\rightarrow\infty}\frac{\ln n}{n}=0[/tex]

    [tex]\sum_{n=1}^{\infty}\frac{1}{ne^0}=\sum_{n=1}^{\infty}\frac{1}{n}[/tex]

    Series diverges.
     
  2. jcsd
  3. Nov 18, 2007 #2

    morphism

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    Are you applying some form of the limit comparison test? If so, then you're right.
     
  4. Nov 18, 2007 #3
    [tex]\lim_{n\rightarrow\infty}\frac{1/ne^{\frac{1}{n}\ln n}}{1/n}=1[/tex]

    so both of them diverge
     
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