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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 9F / iv / Solution

For the sequence in part (iii), part (ii) gives

(\frac{n !}{n ^{n}})^{1/ n} ⟶ e^{- 1} .

Taking reciprocals yields the factorial-over-power series limit

n \to \infty lim \frac{n}{( n ! ) ^{1/ n}} = e .

Solved by gpt-5.6-sol high.

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