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−2n+1n2(n+1)2,for all n≥1,u_{n+1} = u_n - \frac{2 n + 1}{n^2(n + 1)^2}, \quad \textrm{for all } n \geq 1,
and un=1n2.{\displaystyle u_n = \frac{1}{n^2}.}
(i)
Find ∑n=1N2n+1n2(n+1)2.{\displaystyle \sum_{n=1}^N \frac{2 n + 1}{n^2(n + 1)^2}.}
[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=2N2n−1n2(n−1)2.{\displaystyle \sum_{n=2}^N \frac{2 n - 1}{n^2(n - 1)^2}.}
[2]

Answer

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