I am trying to figure out a power series for $(A+\epsilon I)^{-1}$ when $A$ is an invertible matrix and $\epsilon$ is large. The Neumann series can be used when $\epsilon$ is small. Is there something similar for large $\epsilon$?
I am thinking I can use Woodbury matrix identity to convert $(A+\epsilon I)^{-1}$ into a form that requires an inversion of $(I+\epsilon A^{-1})$ instead in which case we can apply the Neumann series. Would this be the best way to do this? Any suggestions?