One thing I could expand on a little is these singularities in Sundman's series. These singularities are nothing more than collisions! It's these collisions and near-collisions that can make the three body problem so hard. Gravity is a 1/r^2 force, so the forces get very strong and change very rapidly when the two objects come close to each other. This is not only hard to handle analytically, but numerically as well!
In this field there is a well known transformation known as the Kustaanheimo-Stiefel transformation (although the full name is not so well known---everyone just refers to it as the KS transformation or KS regularization). What this transformation does is it moves you from a regular Cartesian or polar coordinate system (which has the force diverge when r --> 0) to a coordinate system in which this singularity is moved to infinity. After a KS transformation a computer can integrate even extremely elliptical orbits or collisional orbits. The only downside to this is the algorithmic complexity. In an ordinary Cartesian system, if you were to perform some numerical integration the complexity goes like O(N^2), with N being the number of objects in your system. The complexity of computing an orbit after KS regularization is O(N^3), however. (My understanding of this is that KS regularization works by turning position vectors into quaternions---which then can basically be treated as matrices. Numerically evolving the system then amounts to performing matrix multiplication, which goes as O(N^3). My intuition may be flawed, however!) KS regularization therefore can't help if you have more than ~10 objects. In practice in my research I haven't found it useful even at N=3.
One last thing to mention is that even though these orbits can in principle be solved numerically, they can't in practice. Three body systems are chaotic and the machine precision of your computer leads to round off errors that cause your calculation to diverge from the true solution remarkably quickly. There are special-purpose programs that can calculate orbits to arbitrary precision, but they run really, really slowly (as you might imagine), so they're not useful for more general studies of three body systems. Fortunately, back in 1991, Quinlan & Tremaine showed that there exist so-called "shadow orbits" that track the orbit numerically computed by your computer. These shadow orbits are real orbits for some initial conditions that come arbitrarily close to the numerically calculated orbits. This leads some credence to numerical few body studies. Unfortunately, what's still unknown is whether these shadow orbits have the same statistical properties as the general set of orbits. Not enough research has been done on this question because it's still open and the accuracy of my research depends on this property being true! But ultimately, this question is probably linked to the ergodic hypothesis which is central to the foundations of statistical mechanics---so it's an important question for all physicists, not just me!