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Power series properties proof

  1. Mar 22, 2012 #1
    Hey guys,

    I've been trying to work out this question,

    http://img189.imageshack.us/img189/2954/asdagp.jpg [Broken]

    so the identity theorm is just that if the power series = 0 then the coefficient of the series must be zero.

    Im having trouble seeing how that negative has any influence over the n in the x^n term, to make the x's in powers of either odd or even.

    If you have f(-x) = f(x) then the series would be like

    Ʃa (x-xo)^n = Ʃa (-x-xo)^n = Ʃ(-1)^n a (x+xo)^n

    So how does that make the powers only even? Is there somthing crusial that im missing?

    Thanks
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 22, 2012 #2

    tiny-tim

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    welcome to pf!

    hey linda! welcome to pf! :smile:

    forget xo

    "even" and "odd" mean about x = 0 :wink:

    does that make it easier?
     
  4. Mar 22, 2012 #3
    Thanks!

    Yea that helps, so then

    Ʃa (x)^n = Ʃa (-x)^n = Ʃ(-1)^n a(x)^n

    So is the trick that the only way this can be true is if all the odd powers of n arn't there since the left side will have + a x, + a x^3,.. and the right-a x,- a x^3,... (for an odd ) which can only be true if they are zero an odd = 0?
     
  5. Mar 22, 2012 #4

    tiny-tim

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    yes :smile:

    but it's a lot easier if you combine it into one series …

    ∑ { axn - a(-x)n } = 0 :wink:
     
  6. Mar 22, 2012 #5
    Cool,

    Thanks a lot!
     
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