Cauchy-Hadamard Theorem

Given a Power Series $\sum_{n=1}^{\infty} a_n(z-z_0)^n$, let $$\rho = \frac{1}{\limsup \sqrt[n]{\vert a_n\vert}}.$$ Then the power series Converges Absolutely if $\vert z-z_0\vert<\rho$ and diverges if $\vert z-z_0\vert>\rho$.