Taylor's theorem
Theorem 8.2 (Taylor’s theorem). If \(f\) is analytic on \(\abs {z - z_0} < \rho \), then
\[ f(z) = \sum _{n=0}^{\infty } \frac {f^{(n)}(z_0)}{n!} (z - z_0)^n \]
for every \(z\) in that disc. In particular, the radius of convergence of the Taylor series is at least the distance from \(z_0\) to the nearest point where \(f\) fails to be analytic.
Source: [1, ch. 5]