Taylor's theorem

If $f$ is holomorphic function on an open set and $z_{0}$ is a point in the domain, then there exists a radius $R > 0$ such that

f (z) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (z_{0})}{n!} (z - z_{0})^{n} for all | z - z_{0} | < R .

That is:

The series is not merely formal. It converges.
It converges uniformly to the function.
The radius $R$ is not necessarily maximal; the maximal radius of convergence is determined by the nearest singularity of $f$ .
From here we have that holomorphic function implies analytic function.