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Math Help - 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:


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

    - A sequence is then given such as 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:

     \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:  \frac{1}{x^2 + 1}
    Write:

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

    and use:

    \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 [LaTeX ERROR: Convert failed] as [LaTeX ERROR: Convert failed] 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
    <br />
a_n=n^2.<br />

    <br />
O(n^{\alpha})<br />
    means that

    <br />
\displaystyle { \lim \frac{a_n}{n^{\alpha}}=const<br />
}<br />

    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 [LaTeX ERROR: Convert failed] as [LaTeX ERROR: Convert failed] 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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