Can a Power Series tell when to stop?

Solution 1:

There is a simple criterion, due to Euler. Suppose that your function is known to be analytic in the unit disk. Expand it into the Taylor series at the point $e^{i\theta}/2$. The radius of convergence of the resulting series is $1/2$ if and only if $e^{i\theta}$ is a singularity.

As the radius of convergence is simply $1/\limsup_{n\to\infty}|f^{(n)}(e^{i\theta}/2)/n!|^{1/n}$ this certainly satisfies your requirement: it is in terms of the "behaviour inside the domain" as $z$ approaches singularity. It does not even have to approach too much:-)

This looks trivial, but actually you can extract much from this simple criterion, for example Pringsheim's theorem that a series with non-negative coefficients has a singularity at the point where the circle of convergence intersects the positive ray, or Hadamard's gap theorem. See L. Bieberbach, Analytische Fortsetzung, Springer 1955.