Question 0510

Sigma Notation
2010 Paper 2 Question 2 Variant

Question

(a)
Prove by the method of differences that
r=1n1(r+2)(r+4)=72412n+612n+8.\sum_{r=1}^n \frac{1}{(r + 2) (r + 4)} = \frac{7}{24} - \frac{ 1 }{ 2 n + 6 } - \frac{ 1 }{ 2 n + 8 }.
[4]
(b)
Explain why r=11(r+2)(r+4){\displaystyle \sum_{r=1}^\infty \frac{1}{(r + 2) (r + 4)}} is a convergent series, and state the value of the sum to infinity.
[2]

Answer