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Math Help - series convergence and divergence

  1. #1
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    series convergence and divergence

    hey first of all i d like to start this problem off with saying i have no idea what the ! means so i cant even start this problem.. even if the ! wasnt there i d assume this was a geometric series since its in that section of the book. Other then that i know if it converges i need to find the sum. any help appreciated.

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  2. #2
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    n! ... is read "n" factorial.

    e.g.

    5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1

    n! = n(n-1)(n-2)(n-3)...(3)(2)(1)


    familiar with the ratio test for convergence/divergence ?
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Legendsn3verdie View Post
    hey first of all i d like to start this problem off with saying i have no idea what the ! means so i cant even start this problem.. even if the ! wasnt there i d assume this was a geometric series since its in that section of the book. Other then that i know if it converges i need to find the sum. any help appreciated.

    \frac{n^n}{n!}=\frac{\overbrace{n\cdot{n}\cdot{n}\  cdots}^{n\text{ number of times}}}{\underbrace{n\cdot(n-1)\cdots}_{n\text{ number of times}}}=\frac{n}{n}\cdot\frac{n}{n-1}\cdots\geqslant{1\cdot{1}\cdots=1}

    So we can see that

    \sum_{n=1}^{\infty}1\to\infty\leqslant\sum_{n=1}^{  \infty}\frac{n^n}{n!}
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  4. #4
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    \sum_{n=1}^{\infty}\frac{n^{n}}{n!}

    The ratio test may be a decent choice here.

    {\rho}=\lim_{n\to +\infty}\frac{u_{n+1}}{u_{n}}=\lim_{n\to +\infty}\frac{(n+1)^{n+1}}{(n+1)!}\cdot\frac{n!}{n  ^{n}}

    =\lim_{n\to +\infty}\frac{(n+1)^{n}}{n^{n}}

    =\lim_{n\to +\infty}\left(1+\frac{1}{n}\right)^{n}=e

    Since e>1, then it diverges.

    If you do not recognize it, that last limit is a very famous one. Therefore, one does not have to bother proving it every time.
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