| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2009 |
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| Session | June |
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| Topic | Reduction Formulae |
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7 Let
$$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$
where \(n \geqslant 0\). Show that, for all \(n \geqslant 1\),
$$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 }$$
Hence prove by induction that, for all positive integers \(n\),
$$I _ { n } < n ! .$$