Proof of series representation of inverse of sum of matrices

Question

Is there a proof that $(A+B)^{-1} = A^{-1} - A^{-1}BA^{-1} + A^{-1}BA^{-1}BA^{-1} - \cdots$?

Answer 1

There is no proof because of the following counter example:

Take

$A=\left[\begin{matrix} 1&0\\ 0&0\\ \end{matrix}\right]$

$B=\left[\begin{matrix} 0&0\\ 0&1\\ \end{matrix}\right]$

We have $A+B= \left[\begin{matrix} 1&0\\ 0&1\\ \end{matrix}\right]$ $\Rightarrow$ $(A+B)^{-1}= \left[\begin{matrix} 1&0\\ 0&1\\ \end{matrix}\right]$

But $A$ is not invertible, so $A^{-1}$ is not defined.