The exercise asks me to show that (these are all real matrices) the following relation holds:
$$ (A+B)^{-1} - A^{-1} = A^{-1}\sum_{k=1}^\infty (BA^{-1})^k$$
I suspect that the question is wrong, since the sum should actually read: $\sum_{k=1}^\infty (-1)^k (BA^{-1})^k$ Justification for stating so: $$ (A+B)^{-1} = [A(I+BA^{-1})]^{-1} = (I+BA^{-1})^{-1}A^{-1}\underbrace{=}_{*}A^{-1}\sum_{\mathbf{k=0}}^\infty (-1)^k (BA^{-1})^k=\\A^{-1}+A^{-1}\sum_{k=1}^\infty (BA^{-1})^k \\\implies (A+B)^{-1} - A^{-1} = \sum_{k=1}^\infty (BA^{-1})^k $$
At the step $*$, I suppose I can't commute. Am I right? Thanks in advance.