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Thread: Sequences and series.

  1. #1
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    Sequences and series.

    Hey there.

    I'm wondering if anyone here will be able to assist with the following issues:

    1. I'm currently working on questions asking me to:


    - $\displaystyle Find \: \alpha \: so \: that \: a_n \: = \: O(n^\alpha)$

    - A sequence is then given such as $\displaystyle n^2 \: + \: n \: + \:1 $

    I'm curious as to what type of questions these are, they are listed under "Order of magnitude" in my notes but searching for this returns no useful resources.

    2. MacLaurin Series

    -Lastly, I'm struggling in creating the MacLaurin Series for the function:

    $\displaystyle \frac{1}{x^2 + 1} $

    -I have tried differentiating but this becomes quite difficult after the third derivative or so. I am guessing there is another method to solve this?


    Any help concerning either of these issues would be greatly appreciated.

    Thank you.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by track02 View Post
    -Lastly, I'm struggling in creating the MacLaurin Series for the function: $\displaystyle \frac{1}{x^2 + 1} $
    Write:

    $\displaystyle f(x)=\dfrac{1}{1+x^2}=\dfrac{1}{1-(-x^2)}$

    and use:

    $\displaystyle \displaystyle\sum_{n=0}^{+\infty}t^n=\dfrac{1}{1-t}\quad (|t|<1)$

    Regards.

    Fernando Revilla
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  3. #3
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    Thanks for the assistance.

    Correct me if I am wrong, but by treating $\displaystyle x^2 $ as $\displaystyle (-x^2) $ you are allowing the use of sigma notation for an infinite geometric series?

    Which will have the same effect as using a MacLaurin Series?

    Also, does anyone have any idea where I can find some information regarding my first issue, I've been searching and still nothing!

    Thanks again.
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  4. #4
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    For large n we have
    $\displaystyle
    a_n=n^2.
    $

    $\displaystyle
    O(n^{\alpha})
    $
    means that

    $\displaystyle
    \displaystyle { \lim \frac{a_n}{n^{\alpha}}=const
    }
    $

    that is that these functions have the same order of magnitude.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by track02 View Post
    Correct me if I am wrong, but by treating $\displaystyle x^2 $ as $\displaystyle (-x^2) $ you are allowing the use of sigma notation for an infinite geometric series?
    Right.

    Which will have the same effect as using a MacLaurin Series?
    Right, as a consequence of unicity of the Taylor expansion.

    Regards.

    Fernando Revilla
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