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Homework Help: Power Series- radius of convergence

  1. Apr 22, 2010 #1
    1. The problem statement, all variables and given/known data
    determine the radius of convergence of the given power series

    [tex]\sum[/tex][tex]^{inf}_{n=1}[/tex][tex]\frac{n!x^n}{n^n}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    I did the ratio test
    then I had to take the 'ln'
    but, my answer is this
    |e|<1 for the series to converge.
    It never happens but according to the answers the radius is 'e'.
     
  2. jcsd
  3. Apr 22, 2010 #2
    oh I think I got it, I didn't have to lhopital lnx in the middle ....
    :\
     
  4. Apr 22, 2010 #3
    Applying the Ratio Test, we have the sequence

    [tex]\frac{(n+1)n^n}{(n+1)^{(n+1)}} = \left(\frac{n}{n+1}\right)^n = \left(\frac{1}{1 + \frac{1}{n}}\right)^n[/tex]

    Taking the limit as n goes to infinity... this might look like a familiar limit. Then recall that the Radius of convergence is the reciprocal of this limit.
     
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