Worked Example 5

Express \(\frac{{{r}^{2}}+7r+11}{\left( r+4 \right)!}\) in the form \(\frac{A}{\left( r+2 \right)!}+\frac{B}{\left( r+4 \right)!}\) where \(A\) and \(B\) are constants to be found.

Hence show that \(\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}+7r+11}{\left( r+4 \right)!}=\frac{5}{24}-\frac{n+5}{\left( n+4 \right)!}}\).