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Math Help - Convergence

  1. #1
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    Convergence

    Hi,

    I just want to make sure I've done this right.... I have to find out if this diverges or converges, so I used the ratio test and showed the limit is 1/3, meaning it converges.

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

    I'm not sure how I should do this one...

     \sum_{n=1}^{\infty} ne^{-n}

    Thanks
    Last edited by Benno_11; May 31st 2009 at 01:56 AM. Reason: wrote wrong expression
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  2. #2
    Moo
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    P(I'm here)=1/3, P(I'm there)=t+1/3
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    Hello,
    Quote Originally Posted by Benno_11 View Post
    Hi,

    I just want to make sure I've done this right.... I have to find out if this diverges or converges, so I used the ratio test and showed the limit is 1/3, meaning it converges.

     \sum_{n=1}^{\infty} \frac{n!}{n^n}
    How did you find 1/3 ?

    \frac{a_{n+1}}{a_n}=\dots=(n+1)\cdot\frac{n^n}{(n+  1)^{n+1}}=\left(\frac{n}{n+1}\right)^n=\left(1-\frac 1n\right)^n

    Then remember that \lim_{n\to\infty}\left(1+\frac an\right)^n=e^a

    So here, the limit is e^{-1}=\frac 1e<1

    I'm not sure how I should do this one...

     \sum_{n=1}^{\infty} ne^{-n}

    Thanks
    I'm sure that this writing may enlighten you :
    e^{-n}=(e^{-1})^n
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  3. #3
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    haha, what I meant to write for the first one was
     \sum_{n=1}^{\infty} \frac{n^3}{3^n},

    Thanks for the help.
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