Prove that $\{ \sum_{n=0}^k z^n \}_{k=0}^\infty$ does not converge uniformly? So as the title says, can anyone prove that $\{ \sum_{n=0}^k z^n \}_{k=0}^\infty$ does not converge uniformly 0n the disk $D(0,1)$? I think it would converge uniformly to $1/(1-z)$ since it is a geometric series, but professor posed the problem so I'm thinking that must not be correct. Thoughts?

Realize that $\sum_{k=0}^{\infty} x = 1/(1-x)$ is unbounded at $x=1$ on $(0,1)$. Also realize that $\sum_{k=0}^n x$ is bounded by $n$ on $(0,1)$. From this, you can conclude that convergence cannot be uniform (do you see how?).
