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Homework Help: Series either converges or diverges

  1. Mar 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine whether the series [tex]\sum_{n=1}^{\infty}{\frac{n!}{n^n}}[/tex] converges or diverges.

    2. Relevant equations
    The Comparison Test
    The Limit Comparison Test

    3. The attempt at a solution
    I know it diverges, and i tried [tex]a_n=\frac{n!}{n^n}[/tex] and [tex]b_n=\frac{1}{n^n}[/tex] for the limit comparison test, but it gave me infinity which is useless. I also tried the comparison test saying [tex]a_n=\frac{n!}{n^n} \ge b_n=\frac{1}{n^n}[/tex] but i don't know how to prove that [tex]\frac{1}{n^n}[/tex] diverges
  2. jcsd
  3. Mar 31, 2009 #2
    Re: Converge/Diverge

    Have you learned of the Ratio Test yet? It would simplify this problem greatly.

    Otherwise, just to show that [tex]\frac{1}{n^{n}}[/tex] diverges, that can be re-written as [tex]n^{-n}[/tex] which would not equate to zero using the nth term test for divergence, therefore it DOES diverge.

    I haven't looked at your other work yet, but if all you needed was to prove that 1/n^n diverges then there ya go!
    Last edited: Apr 1, 2009
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