It is stated in Wikipedia without source that the gravitational $N$-body problem can be in principle solved with Taylor series expansions. The process outlined goes like this:
Consider a system of $N$ particles, with the $i$th particle at position $\mathbf{x}_i(t)$ and having mass $m_i$. Then Newton's equation reads $$ \frac{\mathrm{d}^2 \mathbf{x}_i(t)}{\mathrm{d}t^2 } = G \sum_{k = 1, \; k\neq i}^N m_k \frac{\mathbf{x}_k(t) - \mathbf{x}_i(t)}{\lvert \mathbf{x}_k(t) - \mathbf{x}_i(t) \rvert^3}. $$
Since $\mathbf{x}_i(t_0)$ and $ \mathrm{d}\mathbf{x}_k(t_0) / \mathrm{d}t $ are given as initial conditions, by iteration of the above equation we have $$ \frac{\mathrm{d}^n \mathbf{x}_i(t_0)}{\mathrm{d}t^n } $$ for all $n$. Then a Taylor series for the solution at time $t$ is given by the Taylor expansion with respect to initial time $t_0$. The convergence of this series has been raised in another question on PSE. I will follow up and ask: is this simplistic method regarded as an analytical solution to the gravitational $N$-body problem? Or else, is this method in anyway practical for $N$-body simulations?
