Question 0507

Sigma Notation
2007 Paper 2 Question 2 Variant

Question

A sequence u1,u2,u3,…u_1, \allowbreak u_2, \allowbreak u_3, \ldots is such that
un+1=un−1(n−1)n,for all n≥1,u_{n+1} = u_n - \frac{1}{(n-1)n}, \quad \textrm{for all } n \geq 1,
and un=1n−1.{\displaystyle u_n = \frac{1}{n - 1}.}
(i)
Find ∑n=2N1(n−1)n.{\displaystyle \sum_{n=2}^N \frac{1}{(n-1)n}.}
[2]
(ii)
Give a reason why the series in part (i) is convergent and state the sum to infinity.
[2]
(iii)
Use your answer to part (i) to find ∑n=0N1(n+1)(n+2).{\displaystyle \sum_{n=0}^N \frac{1}{(n + 1)(n + 2)}.}
[2]

Answer

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