Solving ODE with negative expansion power series

Solving ODE with negative expansion power series

I am solving a series of ODE, such that each DE is equal to some degree of
term that I'm expanding to. For instance, one DE is this:
$\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$
$g_{ii}$ are given, from metric, but that's not important. I need to
assume that the solution (since I'm looking for components of $\xi^\mu$,
which is a vector with components $\xi^t,\xi^r,\xi^\phi$) is given with
power series of the form:
$\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, and this is to be seen as
expansion around $1/r$ (expansion around $r=\infty$).
Now when I plug this in the ODE I get this
$\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t}
r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where
$\xi^\mu_{n,i}$
is the derivative with the respect to i-th component.
What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my
$\mathcal{O}(r)$ terms cancel each other out? Or?
I'm kinda stuck :\