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Thread: uniform convergence for a power series

  1. #1
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    uniform convergence for a power series

    Let $\displaystyle \sum_{n=0}^\infty a_n(z-z_o)^n$ be a complex power series that converges uniformly on $\displaystyle \mathbb{C}$.
    Show that there is $\displaystyle {N}\epsilon \mathbb{N} $ such that $\displaystyle a_n = 0$ for all $\displaystyle 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 $\displaystyle z_o$ 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 $\displaystyle a_{N+1} \neq 0$
    then do something with the inequality
    $\displaystyle \mid S_N(x)-S(x) \mid < \varepsilon$
    by using $\displaystyle S_{N+1}(x) = S_N(x) + a_{N+1}$

    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 $\displaystyle \epsilon>0$ and number N such that $\displaystyle |S(x)-S_N(x)|<\epsilon$ for all $\displaystyle z\in\mathbb{C}$. This means that $\displaystyle S(x)-S_N(x)=\sum_{n=N+1}^\infty a_n(z-z_0)^n$ is analytic and bounded on $\displaystyle \mathbb{C}$, thus constant by Liouville. Thus $\displaystyle a_n=0$ for all $\displaystyle n\ge N+1$.
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