\(\dfrac{1}{2!}+\dfrac{5}{3!}+\dfrac{11}{4!}+\dfrac{19}{5!}+...+\dfrac{n^2+n-1}{\left(n+1\right)!}\)
\(=\dfrac{1}{2!}+\dfrac{2^2+2-1}{\left(2+1\right)!}+\dfrac{3^2+3-1}{\left(3+1\right)!}+\dfrac{4^2+4-1}{\left(4+1\right)!}+...+\dfrac{n^2+n-1}{\left(n+1\right)!}\)
\(=\dfrac{1}{2!}+\dfrac{2.\left(2+1\right)-1}{\left(2+1\right)!}+\dfrac{3.\left(3+1\right)-1}{\left(3+1\right)!}+\dfrac{4.\left(4+1\right)-1}{\left(4+1\right)!}+...+\dfrac{n.\left(n+1\right)-1}{\left(n+1\right)!}\)
\(=\dfrac{1}{2!}+\dfrac{1}{1!}-\dfrac{1}{3!}+\dfrac{1}{2!}-\dfrac{1}{4!}+\dfrac{1}{3!}-\dfrac{1}{5!}+...+\dfrac{1}{\left(n-1\right)!}-\dfrac{1}{\left(n+1\right)!}\)
\(=\dfrac{1}{2!}+\left(\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+...+\dfrac{1}{\left(n-1\right)!}\right)-\left(\dfrac{1}{3!}+\dfrac{1}{4!}+\dfrac{1}{5!}+...+\dfrac{1}{\left(n+1\right)!}\right)\)
\(=\dfrac{1}{2!}+\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{n!}-\dfrac{1}{\left(n+1\right)!}\)
\(=2-\dfrac{n+1+1}{\left(n+1\right)!}\)
\(=\dfrac{2\left(n+1\right)!-n-2}{\left(n+1\right)!}\)
