The Impossibility ( or Possibility) of Solving $N$-Body Problem

While the N-body Problem is chaotic, a convergent expansion exists. The 3-Body expansion was found by Sundman in 1912, and the full N-body problem in 1991 by Wang.

However, These expansions are pretty much useless for real problems( millions of terms are required for even short times); you're much better off with a numerical integration.

The history of the 3-Body problem is in itself pretty interesting stuff. Check out June Barrow-Green's book which include a pretty good analysis of all the relevant physics, along with a ripping tale.

One easy way to see this is that the N-body problem can be used, with appropriate potentials, to simulate a classical computer, so that as N becomes large, any algorithm which predicts the future behavior at arbitrarily long times has to be at least as computationally complex as a general cN-bit computer (where c is the number of bits you can usefully code per-particle) . Summation of convergent infinite series also simulates a computer, so that's not a useful interpretation of the word "solve". But any good definition of saying "solve" should mean that you reduced the computational complexity of predicting the future behavior from the present state, which can't be done for a general purpose computer.