Taylor Series reaches closest Singularity

Theorem
Let the singularities of a function be the points at which the function is not analytic.

Let $F$ be a complex function.

Let $F$ be analytic everywhere except at a finite number of singularities.

Let $x_0$ be a point in $\R$ where $F$ is analytic.

Let $R \in \R_{>0}$ be the distance from $x_0$ to the closest singularity of $F$.

Let $f = F {\restriction_{\R}}$ be a real function.

Then:
 * the Taylor series of $f$ about $x_0$ converges to $f$ at every point $x \in \R$ satisfying $\left\lvert{x - x_0}\right\rvert < R$

Proof
We have that $F$ is analytic everywhere except at its singularities.

Also, the distance from $x_0$ to the closest singularity of $F$ is $R$.

Therefore:
 * $F$ is analytic at every point $z \in \C$ satisfying $\left\lvert{z - x_0}\right\rvert < R$

The result follows by Convergence of Taylor Series of Function Analytic on Disk.

Also see

 * Taylor Series of Analytic Function has infinite Radius of Convergence