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Math Help - series with factorial

  1. #1
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    series with factorial

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

    determine whether the series converges or diverges

    help?
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  2. #2
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    Hello, buttonbear!

    Without the Ratio Test, I'm not sure of the proof.


    \sum_{n=1}^{\infty}\frac{n!}{n^n} . Determine whether the series converges or diverges
    The Ratio Test provides some interesting maniputations . . .

    R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{n^n}{n!} \;=\;\frac{(n+1)!}{n!}\cdot\frac{n^n}{(n+1)^{n+1}}

    . . = \;\frac{n+1}{1}\cdot\frac{n^n}{(n+1)^{n+1}} \;=\;\frac{n^n}{(n+1)^n} \;=\;\left(\frac{n}{n+1}\right)^n


    Divide top and bottom of the fraction by n\!:\;\;\left(\frac{1}{1+\frac{1}{n}}\right)^n \;=\;\frac{1}{\left(1 + \frac{1}{n}\right)^n}


    Therefore: . \lim_{n\to\infty}R \;=\;\lim_{n\to\infty}\frac{1}{\left(1 + \frac{1}{n}\right)^n} \;=\;\frac{1}{\lim\left(1 + \frac{1}{n}\right)^n} \;=\;\frac{1}{e} \;<\;1

    . . and the series converges.

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  3. #3
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    For all n \geq 2, we have: \frac{n!}{n^n} = \frac{1 \cdot 2 \cdot 3 \cdots (n-1)n}{n \cdot n \cdot n \cdots n \cdot n} \ \ {\color{red} \leq}\ \ \frac{1}{n} \cdot \frac{2}{n} \cdot 1 \cdot 1 \cdots \cdot 1 = \frac{2}{n^2}

    So by comparison ...
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  4. #4
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    i'm sorry, i just didn't fully understand your post..
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  5. #5
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    I've showed that: \frac{n!}{n^n} < \frac{2}{n^2}

    Since \sum \frac{2}{n^2} is convergent, by the comparison test ... what can you conclude?
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  6. #6
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    i can conclude that... because the larger sequence converges..the other one does too?
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  7. #7
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    i'm just kind of confused as how to how you knew to pick \frac{2}{n^2} for your larger sequence? or can it be any larger sequence?
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  8. #8
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    sorry for bumping up an old thread but how did you know to compare it to 2/n^2 as opposed to 1/n^2??
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  9. #9
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    oh nvm i think i see it, because you cant use 1/n^2, you can only use 2/n^2, cause if you only use the first term it would be 1/n, which is divergent according to the p-series, and so you use n>=2..ahh i see it..
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