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Power Series Representation

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data
    I am asked to find a power series for the function f(x) = 2/(1 - x^2), centered at 0.

    2. Relevant equations



    3. The attempt at a solution
    The only part I can't determine is the interval of convergence. I get stuck on the step |x2| < 1. What am I to do next?
     
  2. jcsd
  3. Oct 27, 2012 #2

    haruspex

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    Are you saying you know the convergence domain is |x2| < 1, and you want to express that as a domain for x? Umm... isn't it obvious?
     
  4. Oct 27, 2012 #3
    Not really, either wise I wouldn't have posed the question in the first place. I thought of taking the square root of both sides of the inequality, but wasn't entirely sure what the result would be.
     
  5. Oct 27, 2012 #4

    haruspex

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    There's a very easy simplification of |x2|. Alternatively, you can expand an inequality like |y| < a (where a > 0) as -a < y < a.
     
  6. Oct 27, 2012 #5
    So:

    -1 < x^2 < 1

    +/- sqrt(-1) < x < +/- sqrt(1)

    +/- i < x < +/- 1
    ?
     
  7. Oct 27, 2012 #6

    haruspex

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    Based on the use of x rather than z, I assumed we were dealing with reals here. Was that a wrong assumption?
     
  8. Oct 28, 2012 #7
    No, this problem does not entertain numbers other than reals. You said that I could expand the inequality involving the absolute, of which I did; henceforth, I solved for x, of which clearly resulted in a rather odd looking inequality, namely [itex] \pm i < x < \pm 1 [/itex]. Was this the solution that you had alluded to as being obvious?
     
    Last edited: Oct 28, 2012
  9. Oct 28, 2012 #8

    haruspex

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    There are two problems with that step.
    When operating with inequalities, you must be particularly careful about changing signs unwittingly. You have assumed sqrt(x^2) = x. The sqrt function is defined to return the principal value, i.e. the non-negative square root. Thus sqrt(x^2) = |x|, and the square roots of x^2 are ±|x|.
    Secondly, it makes no sense to take sqrt(-1) if we're dealing with reals. Isn't the inequality -1 < x^2 somewhat redundant? Can you think of a tighter bound?
     
  10. Oct 28, 2012 #9
    Would it be (-1, 1)?
     
  11. Oct 28, 2012 #10
    Looks good to me!

    Think about it like this: if -1<x<1 then surely x^2<1 right?
     
  12. Oct 29, 2012 #11
    Well, certainly my textbook is wrong for saying the interval of convergence is (1, 1). Thank you both for your help.
     
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