Question 0510

Sigma Notation
2010 Paper 2 Question 2 Variant

Question

(a)
Prove by the method of differences that
r=3n1(r+1)(r+3)=94012n+412n+6.\sum_{r=3}^n \frac{1}{(r + 1) (r + 3)} = \frac{9}{40} - \frac{ 1 }{ 2 n + 4 } - \frac{ 1 }{ 2 n + 6 }.
[4]
(b)
Explain why r=31(r+1)(r+3){\displaystyle \sum_{r=3}^\infty \frac{1}{(r + 1) (r + 3)}} is a convergent series, and state the value of the sum to infinity.
[2]

Answer