Question 0510

Sigma Notation
2010 Paper 2 Question 2 Variant

Question

(a)
Prove by the method of differences that
r=2n1r(r+2)=51212n+212n+4.\sum_{r=2}^n \frac{1}{r (r + 2)} = \frac{5}{12} - \frac{ 1 }{ 2 n + 2 } - \frac{ 1 }{ 2 n + 4 }.
[4]
(b)
Explain why r=21r(r+2){\displaystyle \sum_{r=2}^\infty \frac{1}{r (r + 2)}} is a convergent series, and state the value of the sum to infinity.
[2]

Answer