Find where the series $∑^∞_{n=0}(\frac{z}{1+z})^n$ converges uniformly

Question

I was able to find that this series converges when $-\frac{1}{2}<\Re(z)$ by using the fact that we must have that $\limsup_{n\to \infty}\sqrt[n]{|a_n|}<1$. Let $A=\{z\in\mathbb{C}|-\frac{1}{2}<\Re(z)\}$.

What's let to find is that if the series converges uniformly here, and if the series converge uniformly on compact subsets of A?

I am having trouble finding this and any help is appreciated.

Answer 1

Hint: $\sum \zeta^{n}$ converges in $|\zeta | <1$ and uniformly on compact subsets of the unit disk. Equivalently it converges uniformly for $|\zeta| \leq 1-r$ for any $r \in (0,1)$. So you only have to put $\zeta =\frac z {1+z}$ in this.