| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2003 |
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| Session | November |
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| Topic | Sequences and series, recurrence and convergence |
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2 Given that
$$u _ { n } = \frac { 1 } { n ^ { 2 } - n + 1 } - \frac { 1 } { n ^ { 2 } + n + 1 } ,$$
find \(S _ { N } = \sum _ { n = N + 1 } ^ { 2 N } u _ { n }\) in terms of \(N\).
Find a number \(M\) such that \(S _ { N } < 10 ^ { - 20 }\) for all \(N > M\).