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I need help understanding this problem.
D is an open disc at 0. Suppose, for each n, that
on D. Also, on compact subsets of D, where this latter convergence is uniform. If we denote , show that .So the second representation of F(z) is meant to be a power series? If so, I'm basically supposed to show that the coefficients of this power series is given as above? Before I say anything else, keep in mind I'm talking entirely out of my backside. I was pretty ill and missed about a week and a half of lecture for a class I probably don't deserve to be in because I haven't done math literally in about 4 years but this was the only class I could take with my schedule (which is why I'm too intimidated to talk to the prof, and I tried talking to another student but he looked at me like I was mentally handicapped and said he wasn't going to waste his time).
That said, does this problem amount to showing that I can switch the order of summation? If so...huh? I feel lost and helpless, especially because this doesn't seem like a "hard problem". I really just want to know if at least my interpretation is decent and how to go about showing this. I'm feeling pretty hopeless in the course, but need to find a way to at least pass (and nobody at my school tutors this stuff).
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