How to show that this series does not converge uniformly on the open unit disc? Given the series $\sum_{k=0}^\infty z^k $, it is easy to see that it converges locally, but how do I go about showing that it does not also converge uniformly on the open unit disc? I know that for it to converge uniformly on the open disc that $sup{|g(z) - g_k(z)|}$, z element of open unit disc, must equal zero. However, I am finding it difficult to show that this series does not go to zero as k goes to infinity.
Edit:Fixed confusing terminology as mentioned in answer.

Confine attention to real $x$ in the interval $0<x<1$. 
Let 
$$s_n(x)=1+x+x^2+\cdots +x^{n-1}.$$
If we use $s_n(x)$ to approximate the sum, the truncation error is $>x^n$.
Choose a positive $\epsilon$, where for convenience $\epsilon<1$.  We want to make the truncation error $<\epsilon$, so we need 
$$x^n <\epsilon,\qquad \text{or equivalently}\qquad n >\frac{|\ln(\epsilon)|}{|\ln(x)|}.$$
Since $\ln x \to 0$ as $x \to 1^{-}$, the required $n$ grows without bound as $x\to 1^{-}$.
