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.

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