Moving exclamation marks

Calculus Level 3

limnn!!n \large \lim_{n \to \infty} \dfrac{n!}{!n}

The limit above has a closed form. Find this closed form.

Give your answer to 3 decimal places.

Notations:

  • !! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

  • !n!n denotes the number of derangements for nn different objects, !n=n!r=0n(1)rr!\displaystyle !n = n! \sum_{r=0}^n \dfrac{(-1)^r}{r!} .

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