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.
Complex Analysis – How to Show a Series Does Not Converge Uniformly on Open Unit Disc
complex-analysis
Best Answer
This is a simple consequence of the fact that each function $S_k:x\mapsto1+x+\cdots+x^k$ is bounded while the limit function $S:x\mapsto1/(1-x)$ is not.
Hence each function $S_k-S$ is unbounded, that is, the sup-norm of $S_k-S$ is infinite, in particular the sequence of the sup-norms does not converge to zero. This last assertion is equivalent to the fact that $(S_k)$ does not converge uniformly to $S$.