Radius of convergence in the context of Analyticity of holomorphic functions

In complex analysis, a complex-valued function ${\displaystyle f}$ of a complex variable ${\displaystyle z}$:

• is said to be holomorphic at a point ${\displaystyle a}$ if it is differentiable at every point within some open disk centered at ${\displaystyle a}$, and
• is said to be analytic at ${\displaystyle a}$ if in some open disk centered at ${\displaystyle a}$ it can be expanded as a convergent power series $${\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}$$ (this implies that the radius of convergence is positive).

One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. Among the corollaries of this theorem are