Question 0509

Sigma Notation
2009 Paper 1 Question 1 Variant

Question

(a)
Show that
1n−1−2n+1n+1=An3−n,\frac{ 1 }{ n - 1 } - \frac{ 2 }{ n } + \frac{ 1 }{ n + 1 } = \frac{A}{n^3-n},
where A{A} is a constant to be found.
[2]
(b)
Hence find ∑r=4n1r3−r.{\displaystyle \sum_{r=4}^n \frac{1}{r^3-r}.}
(There is no need to express your answer as a single algebraic fraction.)
[3]
(c)
Give a reason why the series ∑r=4∞1r3−r{\displaystyle \sum_{r=4}^\infty \frac{1}{r^3-r}} converges, and write down its value.
[2]

Answer

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