2022 AMC 10B Problem 9

The sum \frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!} can be expressed as a-\frac{1}{b!}, where a and b are positive integers. What is a+b?

\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.