If ${\bf A}$ is an invertible matrix, then for ${\bf B}={\bf A}+\delta {\bf A}$ derive a representation of ${\bf B}^{-1}$ as a matrix power series in $\Delta={\bf A}^{-1}\delta{\bf A}$ and show that it converges when $\|\delta{\bf A}\|<\|{\bf A}^{-1}\|^{-1}$ for any proper matrix norm. State the necessary and sufficient condition on $\Delta$ for convergence of the series.
Any help is appreciated, I've been staring at this for quite a while
Hint: Write the real function $$f(x) = \frac{1}{a+x}$$ as a power series, then replace $a$ by the matrix $A$ and $x$ by the perturbation $\delta A$. Check that everything works for matrices.
Warning: Be careful, order of multiplication matters for matrices!