Poles of power series This may be a trivial question, but I haven't been able to find an answer.
Given a power series about $x_0$ $F(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$, how do we find its (complex) poles? What about the degrees of those poles? Is there an explicit formula (analogous to the explicit formula for the radius of convergence) for the location and nature of the poles (and other singularities)?
 A: I would say this is a very difficult thing to do (from which you can deduce that this is not going to be a very satisfactory answer...).
Having just the power series about one point, the radius of convergence tells you the distance from the point of expansion of the first point at which the function fails to be analytic. This essentially divides into poles, essential singularities, branch points, and the dreaded natural boundary (look up lacunary functions for the latter). (I've not seen a theorem stating these are the only options, but they're the only ones we normally mention. Should anyone have a reference for this sort of classification, I'm interested to see it.)
One thing that you can look up is Padé approximants (or on Scholarpædia). (I admit these are a bit of a mystery to me as well, but they have an uncanny ability to put poles in the right places if the function has poles. If there's a branch cut, the process tends to add a line of poles and zeros along it in a direction tangential to the circle of convergence of the series, if I remember correctly.
The corresponding question for zeros is also difficult (Riemann hypothesis... and the Sendov conjecture).
