I have an asymptotic series expansion that I would like to invert of the form
$ t_2 z + t_3 z^2 + t_4 z^3 + \frac{A_1}{z} + \frac{A_2}{z^2} + ... + \frac{A_9}{z^9} + \mathcal{O}(z^{-10}) = w$
i.e. I would like to be able to write $z$ solutions in the form
$ z_{1,2,3} = (\frac{w}{t_4})^{1/3} \big(...\big)$
where the terms in the brackets are a power series with coefficients that are functions of the coefficients of my original equation.
I have tried using InverseFunction and InverseSeries but to no avail. Is this inversion possible in mathematica?