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.

If I take your wording literally, your difficulty might stem from the fact that you confused the supremum going to zero with the series going to zero pointwise. This is precisely the difference between convergence and uniform convergence. The expression $|g(z)-g_k(z)|$ does go to zero for all $z$ on the open unit disc, but the supremum doesn't. (I'm assuming that by $g(z)$ you mean the series and by $g_k(z)$ its $k$-th partial sum; the question should introduce the notation used if it's not obviously standard.)
