The original poster isn't participating. I add the question "what do you mean by pole" because typical (mechanical wave texts) do not use the term and plots are non-polar if we discount sinusoidal behavior as poles (usually referred to in those terms not polar terms imho).

i'm guessing your working from a school book? Using the keyword Taylor series and radius of convergence I assume (ignoring the equation being attempted, which i might call an oscillatory type equation that is very common) you are using Series ODE method and wish mm to expose the radius of convergence. Explained below: the answer is mm will use well planned chosen method and cannot be asked to use a specific one and the ODE library should have such a feature as to expose "what it is doing" and as yet does not.

did you plot it yet? is the solution DSolve gives a general solution, particular solution, singular solution: and if not why not? is the solution a fuction or relation? also importantly how many parameters are required for the general solution (how many initial conditions are required nor more or less than)? it doesn't say all those things does it?

since this is an 3 day old post without replies let me be brief on this reply:

search these forums for past ODE questions. you will find Mathematica uses 3 major ODE solving libraries and that Help has full documentation on these. search past posts if you can't find "the exact expressions mm uses for ode solving" in Help.

Remember that your ODE book said approximation methods are ultimately more important than hand methods (of which series method is one). No. Mathematica cannot be "asked" to hand solve a given equation in each (valid/invalid) symbolic method. It's highly advanced library chooses a best method which may not be series.

No i don't believe the methods leave a trail of data (such as substitutions, temporary D or I results, or radius calculations) that you can get at easily.

You choice, if you persisted (i'm sure you would not), would be to code it step-by-step by hand (well you might do this) or to use the .nb full code as given in Help and alter the code to return you the radius whenever certain series sol'n were made.

You MIGHT try looking in Context[] to see if the differential solver has it's own context and if so - if it has a variable in a context it uses for only that purpose . (for some wolfram library solvers this might turn up what you need: but for what we're talking about i imagine it would be a waste of time to check and find nothing usable)