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Homework Help: Taylor Series using Geometric Series and Power Series

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data
    See figure attached.


    2. Relevant equations



    3. The attempt at a solution

    Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the,

    [tex]\frac{1}{x^{2}}[/tex]

    [tex]\int x^{-2} = \frac{-1}{x} + C[/tex]

    How do I deal with the C?

    If I can get,

    [tex]\frac{-1}{x}[/tex]

    I can work with it to get something like the following,

    [tex]\frac{\text{first term of geometric series}}{1 - \text{common ratio}}[/tex]

    So what do I do about the C? Once I figure this out I can make more of an attempt into shaping,

    [tex] \frac{-1}{x}[/tex]

    into the form mentioned above.

    Any ideas?

    Thanks again!
     

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    Last edited: Sep 29, 2010
  2. jcsd
  3. Sep 30, 2010 #2

    vela

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    You're going about it backwards. Use

    [tex]\frac{1}{x^2} = -\frac{d}{dx}\left(\frac{1}{x}\right)[/tex]
     
  4. Sep 30, 2010 #3
    Alrighty I think I've got a series for,

    [tex]\frac{1}{x^{2}}[/tex]

    See figure attached. Is this correct?

    I can't seem to figure out how to express it sigma notation however.

    Any ideas?
     

    Attached Files:

  5. Sep 30, 2010 #4

    vela

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    No, it looks like you integrated the series, but you want to differentiate -1/x to get 1/x2.
     
  6. Sep 30, 2010 #5
    Whoops!

    How does this look? (See figure attached)
     

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  7. Sep 30, 2010 #6

    vela

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    Looks good!
     
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