10. Theorem 2.1.1

Let $f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n}$ be a power series with radius of convergence $R = \frac{1}{\underset{n \to \infty}{lim sup} \sqrt[n]{| a_{n} |}}$ . Then $f$ is holomorphic on $B_{R} (z_{0})$ , and it has derivative
:?:

\sum_{n = 1}^{\infty} n a_{n} (z - z_{0})^{n - 1}