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Another question

  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}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p[/tex] where p is a parameter[/tex]

    3. The attempt at a solution


    so by using Root Test i decided that

    Which gives that series converges
  2. jcsd
  3. Nov 18, 2007 #2


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    What if p=0, for example?
  4. Nov 18, 2007 #3
    Oh,I see.Limit is 1.I must try another way.
  5. Nov 18, 2007 #4
    [tex]\sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\ln \left(1+\frac{1}{n}\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\left(\frac{1}{n}-\frac{1}{2n^2}+O(\frac{1}{n^3})\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{1-\frac{1}{2n}+O(\frac{1}{n^2})}\right)^p[/tex]



    Is it converges for all p<1?
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