Powerful Limit

Calculus Level 5

\[\large f(n) = \sum_{r=0}^{n} \dbinom{n}{r} r^r (2-r)^{n-r} \]

Find the value of the closed form of \(\displaystyle \text{L} = \lim_{n\to\infty} \dfrac{f(n)}{n!} \).

Submit your answer as \( \lfloor 1000 \text{L} \rfloor \).

You may use a calculator for the final step of your calculation.

Details and assumptions:

  • Assume that we adopt the convention that \(0^0 = 1\).
  • \( \lfloor \cdot \rfloor \) denotes the floor function.

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