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Math Help - uniform convergence for a power series

  1. #1
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    uniform convergence for a power series

    Let  \sum_{n=0}^\infty a_n(z-z_o)^n be a complex power series that converges uniformly on \mathbb{C}.
    Show that there is {N}\epsilon \mathbb{N} such that a_n = 0 for all n>N .

    and I'm kind of impressed with myself for learning all the latex to write this out

    Ok so I'm thinking that without loss of generality I can let z_o be 0
    then I'm pretty sure I'd have to use the definition of uniform convergences to create a contradiction
    but I don't know what to do for this
    my first guess was to say that suppose a_{N+1} \neq 0
    then do something with the inequality
    \mid S_N(x)-S(x) \mid < \varepsilon
    by using S_{N+1}(x) = S_N(x) + a_{N+1}

    I think I might be close, but I don't know what to do from here
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  2. #2
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    assume there exist \epsilon>0 and number N such that |S(x)-S_N(x)|<\epsilon for all z\in\mathbb{C}. This means that S(x)-S_N(x)=\sum_{n=N+1}^\infty a_n(z-z_0)^n is analytic and bounded on \mathbb{C}, thus constant by Liouville. Thus a_n=0 for all n\ge N+1.
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