Cauchy–Hadamard
Theorem 5.1 (Cauchy–Hadamard). Let \(\sum a_n (z - z_0)^n\) be a power series and define \(R \in [0, \infty ]\) by
\[ \frac {1}{R} = \limsup _{n \to \infty } \abs {a_n}^{1/n}, \]
with \(1/0 = \infty \) and \(1/\infty = 0\). The series converges absolutely for \(\abs {z - z_0} < R\) and diverges for \(\abs {z - z_0} > R\). We call \(R\) the radius of convergence.
Source: [1, ch. 5]