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Power series

  1. Jun 11, 2007 #1
    I saw that there were a pair of threads that discussed these series but since it was far away from my knowledge I decided to start my own.
    I have just been introduced to these series and found out that the series on the form of:
    a_0 + a_1*x + a_2*x^2 + a_3*x^3 + ....

    a_0 = 1
    a_1 = 0
    a_2 = 1
    a_3 = 0
    a_4 = 1
    Can be descirbed with the equation (1-x^2)^-1 as long as -1 ≤ x ≤ 1, i.e. it is convergence.

    But I also read that even though the answers where x is either smallar than -1 or bigger than 1 seems like nonsens, this isnt the case. For example when x = 2 the author writes
    That the equation,(1-x^2)^-1, gives.
    Now in which way can -1/3 make any meaning when the series clearly goes to infinity?
    Last edited: Jun 11, 2007
  2. jcsd
  3. Jun 11, 2007 #2


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    There is a concept in complex variable theory called analytic continuation. Using this principal a function analytic within a certain domain (in this case |z|<1) can be extened to the rest of the complex plane, with singularities. Here the singularities are at z=-1 and z=1. Since the power series is equivalent to (1-z^2)^-1 for z within the unit circle, the closed form is the correct analytic continuation.
  4. Jun 11, 2007 #3
    Ah, that makes sense ;)
    Thank you.
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