2022 RI Promo Q4

Find \(\sum\limits_{r=1}^{n}{\frac{1}{\left( r+1 \right)\left( r+3 \right)}}\), where \(n\ge 3\). (There is no need to express your answer as a single algebraic fraction.)[5]

Explain why \(\sum\limits_{r=1}^{\infty }{\frac{1}{\left( r+1 \right)\left( r+3 \right)}}\) is a convergent series, and state the value of the sum to infinity.[2]