Question 0510

Sigma Notation
2010 Paper 2 Question 2 Variant

Question

(a)
Prove by the method of differences that
r=2n1(r+3)(r+5)=116012n+812n+10.\sum_{r=2}^n \frac{1}{(r + 3) (r + 5)} = \frac{11}{60} - \frac{ 1 }{ 2 n + 8 } - \frac{ 1 }{ 2 n + 10 }.
[4]
(b)
Explain why r=21(r+3)(r+5){\displaystyle \sum_{r=2}^\infty \frac{1}{(r + 3) (r + 5)}} is a convergent series, and state the value of the sum to infinity.
[2]

Answer