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.