There is the question and answer about it there: Asymptotic expansion of the solution of $f_n(x)=x^5+nx-1=0$.
But what is about formal computation?
Suppose that $x = \sum_{k=1}^{\infty} a_k n^{-k}$. Then I can substitute $x$ in the equation by this series and obtain the correct answer. But I have no idea about validity of this computation :(