# Convergence of Power Series

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.