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Analytic on an interval/expressing with a different power series

  1. Apr 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose the real power series [tex]\sum ^{\infty}_{n=0}c_{n}x^{n}[/tex] has radius of convergence R > 0. Define f:= [tex]\sum ^{\infty}_{n=0}c_{n}x^{n}[/tex] on I:= (-R, R) and let b [tex]\in[/tex] I. Show that there exists a power series [tex]\sum d_{n}(x-b)^{n}[/tex] that converges to f(x) for |x-b| < r - |b|.

    2. Relevant equations

    None that I can think of.

    3. The attempt at a solution

    I don't even know where to begin. Obviously the function is analytic on the open interval I because it is defined by a power series that converges on I. Intuitively, I understand that the function can be represented by a power series with a different center (in I) and smaller radius of convergence, but I can't think of how to start demonstrating this rigorously.
     
    Last edited: Apr 1, 2009
  2. jcsd
  3. Apr 2, 2009 #2

    HallsofIvy

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    If f(x) is as defined, what can you say about f(u) where u= x+ b? Notice that x= u- b.
     
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