$$?= \sum _ {n=1}^ {2021} \frac{n+2}{n!+(n+1)!+(n+2)!} \Longrightarrow \frac{n+2}{n!+(n+1)!+(n+2)!} = \frac{n+2}{n!+(n+1).n!+(n+2).(n+1).n!} = \frac{n+2}{n!(1+n+1+(n+2).(n+2))} \Longrightarrow \frac{n+2}{n!.(n+2).(n+2)} = \frac{1}{n!(n+2)} = \frac{n+2-1}{(n+2)!} = \frac{1}{(n+1)!} - \frac{1}{(n+2)!} \Longrightarrow \sum _ {n=1}^ {2021} \frac{n+2}{n!+(n+1)!+(n+2)!} = \sum _ {n=1}^ {2021}[ \frac{1}{(n+1)!} - \frac{1}{(n+2)!} ] = \frac{1}{2} - \frac{1}{2023!} $$
