# Thread: uniform convergence for a power series

1. ## 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

2. 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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