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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 continuous also in x=±r, but I'm not sure if that's what I am looking for.
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 continuous also in x=±r, but I'm not sure if that's what I am looking for.