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Radius of Convergence for Moderately Complicated Series

  1. Sep 6, 2009 #1
    1. The problem statement:

    Show that the following series has a radius of convergence equal to [tex] exp\left(-p\right) [/tex]

    2. Relevant equations

    For p real:

    [tex]\Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n}[/tex]

    3. The attempt at a solution
    [tex]\stackrel{lim}{n\rightarrow\infty}\left|a_{n}\right|^{1/n} = \frac{1}{R} = \left(\frac{n+p}{n}\right)^{n}
    =exp\left(n\left(ln\left(\frac{n+p}{n}\right)\right)\right)[/tex]

    Apart from playing with the logarithm after that I cannot seem to reach the required answer.
    Any help would be greatly appreciated.
     
  2. jcsd
  3. Sep 6, 2009 #2
    What's the limit definition of the exponential function?
     
  4. Sep 6, 2009 #3
    [tex]exp\left(-p\right) = e^{\left(-p)\right}[/tex]

    is that what you meant?
     
  5. Sep 6, 2009 #4
    Do you know this limit:
    [tex]\lim_{n\to\infty}\left(1+\frac{p}{n}\right)^n[/tex]
     
  6. Sep 6, 2009 #5
    Completely overlooked that! Thanks very much.
     
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