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How to prove the convergence

  1. Jan 17, 2005 #1
    Hi all,

    suppose the following sum:

    \sum_{n = 1}^{\infty} \frac{n!}{n^{n}}

    I had a feeling it should converge, but I can't find a way how to prove that. I tried d'Alembert's criterion:

    \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} < 1 \Rightarrow \sum a_{n} converges

    But the limit is 1 so it doesn't give anything. In fact, if I were able to prove that

    \lim_{n \rightarrow \infty} \sqrt[n]{n!} = 1

    I would have it. But I don't know how to make a prove of that...Which criterion or rule should I use here?

    Thank you.
  2. jcsd
  3. Jan 17, 2005 #2
    Whoops, the limit isn't 1, it is infinity....
  4. Jan 17, 2005 #3


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    Check d'Alembert again.
    [itex]\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_n}|[/itex] is smaller than 1.

    Hint: Make use of: [itex]\left(1+\frac{1}{n}\right)^n \to e[/itex] as [itex]n \to \infty[/itex]
  5. Jan 18, 2005 #4
    Thank you Galileo, now I can see it goes to 1/e.
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