Russian Book Limit

Calculus Level 3

Calculate lim ⁡ n → ∞ n k C n k \lim_{n \to \infty}\frac{n^{k}}{C^{k}_{n}} where k ∈ N k \in \mathbf{N} and C n k = ( n k ) C^{k}_{n}=\binom{n}{k}

k(k-1) k! (k-1)! k(k+1)

Cid Moraes by Cid Moraes , 35, Brazil

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2 solutions

Lu Chee Ket
Jan 24, 2015

I applied Excel to verify. Good to know this.

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thank you!

Cid Moraes - 6 years, 4 months ago
Cid Moraes
Jan 24, 2015

It's clear that n k ( n k ) = n k ⋅ k ! n ( n − 1 ) ( n − 2 ) ⋯ ( n − k + 1 ) \frac{n^{k}}{\binom{n}{k}}=\frac{n^{k}\cdot k!}{n(n-1)(n-2)\cdots(n-k+1)} = k ! 1 ⋅ ( 1 − 1 n ) ( 1 − 2 n ) ⋯ ( 1 − k − 1 n ) → k ! =\frac{k!}{1\cdot\left(1-\frac{1}n\right)\left(1-\frac{2}n\right)\cdots\left(1-\frac{k-1}{n}\right)} \to k! as n → ∞ n \to \infty

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