Heuristically, "expand" \begin{align*} \frac{1}{I+B}&=I-B+B^2+\cdots+(-1)^{k-1}B^{k-1}+(-1)^kB^k+\cdots\\ &=I-B+B^2+\cdots+(-1)^{k-1}B^{k-1}. \end{align*} To have a rigorous solution, verify directly that $$ (I+B)(I-B+B^2+\cdots+(-1)^{k-1}B^{k-1})=I. $$

Heuristically, "expand" \begin{align*} \frac{I}{I+B}&=I-B+B^2+\cdots+(-1)^{k-1}B^{k-1}+(-1)^kB^k+\cdots\\ &=I-B+B^2+\cdots+(-1)^{k-1}B^{k-1}. \end{align*} To have a rigorous solution, verify directly that $$ (I+B)(I-B+B^2+\cdots+(-1)^{k-1}B^{k-1})=I. $$