2020 DHS P1 Q8

Show that \(\frac{2x+9}{x(x+2)(x+3)}=\frac{3}{2}\left( \frac{1}{x}-\frac{1}{x+2} \right)+\left( \frac{1}{x+3}-\frac{1}{x+2} \right)\).[1]

Using the method of differences, show that

\(\sum\limits_{r=1}^{n}{\frac{2r+9}{r(r+2)(r+3)}}=\frac{23}{12}-\frac{3}{2}\left( \frac{1}{n+1} \right)-\frac{3}{2}\left( \frac{1}{n+2} \right)+\frac{1}{n+3}\).[4]

Explain why \(\sum\limits_{r=1}^{\infty }{\frac{2r+9}{r(r+2)(r+3)}}\) is convergent and find its exact value.[2]

Using the result in part (ii), deduce \(\sum\limits_{r=3}^{n}{\frac{2r+7}{(r+2)({{r}^{2}}-1)}}\).[3]