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Power series representation

  1. Feb 2, 2013 #1
    1. The problem statement, all variables and given/known data
    write a power series representation of the following:
    [tex] \frac{x}{15x^2 +1} [/tex]


    2. Relevant equations
    the formula
    [tex] \frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n [/tex]


    3. The attempt at a solution
    we can rewrite the summnd like
    [tex] \frac{x}{15} \left( \frac{1}{1+\frac{x^2}{15}} \right)[/tex]
    we can write the denominator from the above term as:

    [tex] 1 - \left( - \left( \frac{x}{\sqrt{15}} \right)^2 \right) [/tex]

    so using the above term we can write the series like:
    [tex] \frac{x}{15} \sum_{n=0}^∞ (-1)^n \frac{x^{2n}}{15^{n/2}}[/tex] /known data[/b]

    and this simplifies to:

    [tex] \sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{15^{n/2 + 1}} [/tex]

    is that correct? This is the basis for the second part which asks for the interval of convergence
    I cant write absolute value, but here goes:
    [tex] \frac{x^2}{\sqrt{15}} < 1[/tex]

    [tex] x < \sqrt{\sqrt{15}} [/tex]

    This means that the interval is

    [tex] \left( -15^{1/4} , 15^{1/4} \right) [/tex]
    Unfortunately I m getting the answer wrong as per the computer... can you please take a look and see if this is correct or not?

    Thank you for your help. It is greatly appreciated!
     
  2. jcsd
  3. Feb 2, 2013 #2

    vela

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    This isn't correct. Try multiplying out the denominator to see this.
     
  4. Feb 2, 2013 #3
    You're right... my bad

    the term should go

    [tex] x \frac{1}{1-(-15x^2)} [/tex]
    which is
    [tex] x \sum (-15x^2)^n [/tex]
    [tex] \sum (-1)^n 15^n x^{2n+1} [/tex]

    is that correct?
    And it follows that:
    [tex] \left| -15x^2 \right| < 1 [/tex]
    and solving this we get
    [tex] \left| x^2 \right| < \frac{1}{\sqrt{15}} [/tex]

    Is this correct?
     
  5. Feb 3, 2013 #4
    Hey can you let me know if this is corrct what I did?

    Thank you for your help
     
  6. Feb 3, 2013 #5

    vela

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    Looks good except for the typo in your last line. It should be |x| not |x2|.
     
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