AMC 10 2022b 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 $a+b$? $\textbf{(A)}\ 2020 \textbf{(B)}\ 2021 \textbf{(C)}\ 2022 \textbf{(D)}\ 2023 \textbf{(E)}\ 2024$