Cauchy-Hadamard Theorem

The Radius of Convergence of the Taylor Series

$$a_0+a_1z+a_2z^2+\ldots$$

is

$$r={1\over\overline{\lim\limits_{n\to\infty}} (\vert a_n\vert)^{1/n}}.$$

See also Radius of Convergence, Taylor Series