Here we study the function
\[
z\mapsto{1-z^n \over 1-z}
\]
For natural numbers $n$ this corresponds to the finite sum
\[
1+z+z^2+z^2+\cdots+z^{n-1}.
\]
This finite is closely related to the infinite series
\[
1+z+z^2+z^2+z^3+\cdots
\]
This series converges for values of $z$ in the unit circle.
In fact if $n\to\infty$ this series converges to
\[
{1\over 1-z}.
\]
In can be seen how inside the circle of convergence this function stabilizes to this function,
while outside is simply diverges.
Amazingly the behavior is reversed if $n$ is chosen to be negative.

angles
radii
linear diagram