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Thread: convergence proof

  1. #1
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    convergence proof

    Prove that the sequence $\displaystyle a_{n} = \frac{n^3}{n!} $ converges.

    Intuitively this seems to converge to $\displaystyle 0 $ (e.g. its $\displaystyle o(n) $). So for all $\displaystyle \varepsilon > 0 $ there is an $\displaystyle N \in \mathbb{N} $, such that for $\displaystyle n \geq N $, $\displaystyle |a_n| < \varepsilon $.

    Is that the end of the proof? Would you choose an $\displaystyle N \in \mathbb{N} $ such that $\displaystyle \frac{N^3}{N!} < \varepsilon $?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by particlejohn View Post
    Prove that the sequence $\displaystyle a_{n} = \frac{n^3}{n!} $ converges.

    Intuitively this seems to converge to $\displaystyle 0 $ (e.g. its $\displaystyle o(n) $). So for all $\displaystyle \varepsilon > 0 $ there is an $\displaystyle N \in \mathbb{N} $, such that for $\displaystyle n \geq N $, $\displaystyle |a_n| < \varepsilon $.

    Is that the end of the proof? Would you choose an $\displaystyle N \in \mathbb{N} $ such that $\displaystyle \frac{N^3}{N!} < \varepsilon $?
    Consider the series $\displaystyle \sum a_n$. note that it converges by the ratio test. thus, $\displaystyle \lim a_n = 0$ by the theorem $\displaystyle \sum b_n \text{ converges} \implies \lim b_n = 0$ (i forgot what it's called ), and hence $\displaystyle a_n$ converges (to 0).
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  3. #3
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Jhevon View Post
    C theorem $\displaystyle \sum b_n \text{ converges} \implies \lim b_n = 0$ (i forgot what it's called ),
    contrapositive of the divergence test..

    ratio test: $\displaystyle \lim \frac{a_{n+1}}{a_n} = \lim\frac{(n+1)^3}{(n+1)!} \cdot \frac{n!}{n^3} = \lim\frac{(n+1)^3}{(n+1)} \cdot \frac{1}{n^3} = \lim\frac{(n+1)^2}{n^3} =0 < 1$
    Last edited by kalagota; Jul 6th 2008 at 07:22 PM. Reason: i figure out there was a mistake on my computation..
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kalagota View Post
    contrapositive of the divergence test..
    haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic
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  5. #5
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Jhevon View Post
    haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic
    It make sense..

    it was a remark of the divergence test when we discussed it before.. haha

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kalagota View Post
    It make sense..

    it was a remark of the divergence test when we discussed it before.. haha
    haha, yup

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    i answered you in a PM
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  7. #7
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Jhevon View Post
    i answered you in a PM
    yes, and i am fixing it already.. thanks!
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