Proof of asymptotic expansion of binomial coefficient

Question

here's the problem I'm currently stuck on:

Prove that (for $k$ fixed):

$$\binom{N}{k}=\frac{N^{k}}{k!}+O(N^{k-1})$$

I know that:

$$\binom{N}{k}\le\frac{N^{k}}{k!}$$

But I'm not sure how to rigorously acquire that $O(N^{r-1})$ term. Any help? Question is exercise 4.4 of Sedgewick & Flagolet's "Analysis of Algorithms", 2nd Edition, if that helps.

score 4 · Accepted Answer · answered Apr 20 '13 at 18:08

4

Just expand the definition of the binomial coefficient in terms of factorials: $$\binom{N}{k} = \frac{N(N-1)(N-2)\cdots(N-k+1)}{k!} = \frac{N^k + aN^{k-1} + bN^{k-2} + \cdots}{k!}$$ since the numerator is just a polynomial.

answered Apr 20 '13 at 18:08

not all wrong

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facepalm. I knew it was going to be something ridiculously simple that I was just overthinking. Thanks! – Jeremy Holland Apr 20 '13 at 18:50
Give me an upvote and all is forgiven ;) It's fine to have blank moments, we all get them. A combinatoric proof would be fun. Should be possible with just s little more thought! – not all wrong Apr 20 '13 at 19:20
Happily will as soon as I have the rep. Just joined today. :) – Jeremy Holland Apr 20 '13 at 19:21
It was a joke, don't worry about it! Welcome, and thanks for asking a question well! (: – not all wrong Apr 20 '13 at 19:33

Proof of asymptotic expansion of binomial coefficient

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