N2010 P2 Q2

N2010 P2 Q2

Junior College 2
11 marks
  1. Prove by mathematical induction that \(\sum_{r=1}^n r(r+2)=\frac16n(n+1)(2n+7)\).[5]
    1. Prove by the method of differences that \(\sum_{r=1}^n\frac1{r(r+2)}=\frac34-\frac1{2(n+1)}-\frac1{2(n+2)}\).[4]
    2. Explain why \(\sum_{r=1}^{\infty}\frac1{r(r+2)}\) is a convergent series, and state the value of the sum to infinity.[2]

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Answer:(i) Induction proved. (ii) Telescoping formula proved; sum to infinity \(3/4\).

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