Let $M(\epsilon)$ be the square matrix $M(\epsilon) = A + \epsilon B$ where $A,B$ are square matrices. Throughout this post, suppose that $M(\epsilon)$ is invertible for arbitrarily small but positive $\epsilon$, but $M(0) = A$ is not invertible (note that my question becomes very easy when $A$ is invertible by just using the Neumann series of the inverse). I'm interested in finding the limiting behavior of the inverse $M(\epsilon)^{-1}$ as $\epsilon\to 0^+$. Specifically, I'd like to expand it as $$M(\epsilon)^{-1} = \sum_{n=-\infty}^\infty D_n \epsilon^n,$$ where $D_n$ are some matrices. How can I find the matrices $D_n$?
Ultimately, I am interested in the limiting behavior of the solution $x$ to the equation $M(\epsilon)x = c$, where $x,c$ are vectors. Specifically, suppose that $x(\epsilon)$ is the solution to $M(\epsilon)x(\epsilon) = c$. I'm interested in knowing $$\lim_{\epsilon\to 0^+} \frac{x(\epsilon)}{\lVert x(\epsilon) \rVert_{\infty}} = \lim_{\epsilon\to 0^+} \frac{M(\epsilon)^{-1}c}{\lVert M(\epsilon)^{-1}c \rVert_{\infty}}.$$
If $M(\epsilon)^{-1} = \sum_{n=s}^\infty D_n \epsilon^n$ for some finite $s\in\mathbb Z$, then I think it follows that $\lim_{\epsilon\to 0^+} \frac{x(\epsilon)}{\lVert x(\epsilon) \rVert_{\infty}} = \frac{D_s c}{\lVert D_s c \rVert_\infty}$. Hence, I want to determine $s$ and $D_s$.