If $\|K\|<1$, the partial <Neumann series> satisfies
$$
(I-K)\sum_{n=0}^NK^n=I-K^{N+1}.
$$
Since $\|K^{N+1}\|\leq\|K\|^{N+1}\to0$, the series converges in operator norm and
$$
\boxed{(I-K)^{-1}=\sum_{n=0}^\infty K^n}.
$$
If $\tau\ne0$ and $\|I-\tau K\|<1$, apply this identity to $I-\tau K$:
$$
\tau K=I-(I-\tau K),
\qquad
\boxed{K^{-1}=\tau\sum_{n=0}^\infty(I-\tau K)^n}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-326/3/b/solution