Question 0510

Sigma Notation
2010 Paper 2 Question 2 Variant

Question

(a)
Prove by the method of differences that
r=2n1(r+2)(r+4)=94012n+612n+8.\sum_{r=2}^n \frac{1}{(r + 2) (r + 4)} = \frac{9}{40} - \frac{ 1 }{ 2 n + 6 } - \frac{ 1 }{ 2 n + 8 }.
[4]
(b)
Explain why r=21(r+2)(r+4){\displaystyle \sum_{r=2}^\infty \frac{1}{(r + 2) (r + 4)}} is a convergent series, and state the value of the sum to infinity.
[2]

Answer