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Power series

  1. Jun 20, 2013 #1
    At exam today I was to calculate an improper integral of a function f defined by a power series.
    The power series had radius of convergence r=1.
    Inside this radius you could of course integrate each term, i.e. symbologically:
    ∫Ʃ = Ʃ∫
    The only problem is that the improper integral went from 0 to 1.
    Is it then true that:
    limx->1[∫Ʃ ]=limx->1[Ʃ∫]
    and what theorem assures this? At the exam I didn't think about this unfortunately, but I would probably not have known what to do anyways. I think there is a theorem called Abels theorem which shows that a power series is continous also in x=±r, but I'm not sure if that's what im looking for.
     
  2. jcsd
  3. Jun 20, 2013 #2

    Zondrina

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    Are you familiar with uniform convergence?

    Abel's theorem basically says that if a series converges at x = R, it converges uniformly over the interval
    0 ≤ x ≤ R.

    As a consequence, you can also observe uniform convergence over the interval -R ≤ x ≤ 0.

    A corollary would be that it converges on -R ≤ x ≤ R.
     
    Last edited: Jun 20, 2013
  4. Jun 20, 2013 #3
    Yes I am familiar with uniform convergence. And you agree that from what I told you I would have had to invoke Abels theorem to give a satisfactory answer?
     
  5. Jun 20, 2013 #4

    Zondrina

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    Uniform convergence would be a sufficient condition for you to be able to switch limits/derivatives/integrals around. So yes, Abel's theorem could be used here as it guarantees the uniform convergence over the whole interval [-R,R].
     
  6. Jun 20, 2013 #5

    Office_Shredder

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    I don't think you need anything fancy at all. Assuming that your variable x is "integrate from 0 to x" then for every x<1, the thing you have written in square brackets [ ] is equal whether the integration or the summation comes first, because you're integrating over [0,x] which is bounded away from 1. You just did algebra inside the limit which is legal for the values of x that you are considering
     
  7. Jun 20, 2013 #6
    Im switching 2 limits:
    limx->1limn->∞[Ʃanxn] = limn->∞limx->1[Ʃanxn]
    I only know that the power series converges for lxl<1
     
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