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Homework Help: Help with sum of infinite series using the root test.

  1. Feb 28, 2010 #1
    The problem statement, all variables and given/known data

    Does it converge or diverge?

    Sum n=0 to infinity : (n/(n+1))^(n^2)

    The attempt at a solution

    I know I need to use the root test.
    But what I get is ...

    Limit to infinity : (n/(n+1))^n

    It seems that the n/(n+1) would go to 1 because when you multiply the top and bottom by n^-1 to get 1/(1+1/n) the 1/n goes to zero and you have 1/1. 1^n is 1, and according to the root test, it is inconclusive. However, the series converges and apparently the limit goes to 1/e, but I'm not sure how.

    Also, does the latex syntax not work in preview mode or something?
     
  2. jcsd
  3. Feb 28, 2010 #2

    vela

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    To evaluate

    [tex]\lim_{n\rightarrow\infty} (1+1/n)^n[/tex]

    try finding the limit of the log of the function.
     
  4. Feb 28, 2010 #3
    I have no idea how.
     
  5. Mar 1, 2010 #4

    HallsofIvy

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    Are you saying that you do not know how to find a logarithm??

    If [itex]y= (1+ 1/n)^n[/itex], then [itex]log(y)= n log(1+ 1/n)[/itex]. What is the limit of that as n goes to infinity? Perhaps L'Hopital's rule or writing log(1+ 1/n) as a Taylor's series will help.
     
  6. Mar 1, 2010 #5
    [tex]\lim_{n\to\infty} \left( \frac{n}{n+1} \right)^n = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^{-n} = ?[/tex]

    Recall:
    [tex]\lim_{n\to\infty} \left( 1 + \frac{a}{n^k} \right)^{bn^k} = e^{ab}[/tex]
     
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