| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2002 |
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| Session | November |
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| Topic | Sequences and series, recurrence and convergence |
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1 Given that
$$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$
find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\).
Hence determine the set of values of \(x\) for which the infinite series
$$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$
is convergent and give the sum to infinity for cases where this exists.