When does the power series converge?

Question

For what values of $z\in \mathbb{C}$ does $$\sum_{n=0}^{\infty} \left( \frac{z}{z+1}\right)^n $$ converge?

Edit: I tried looking at the function $f(z)=1/(z+1)$ to see what values of $z$ might be mapped such that the series converges, but it seemed confusing since I don’t know much complex analysis at this point.

Answer 1

HINT

Recall that $f(z) = \sum_{n=0}^\infty z^n$ converges if and only if $|z| < 1$. Can you use this to characterize the convergence of your series?

Answer 2

Hint: $\require{cancel}\;|z|^2 \lt |z+1|^2 \iff z\bar z \lt (z+1)(\bar z +1) = z \bar z + z + \bar z + 1 \iff z+ \bar z \gt -1\,$.