Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

Asked Nov 02 '14 at 14:40

Active Nov 02 '14 at 15:31

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We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help?

Note: I want to show it is equal to $-p\log{p}-(1-p)\log{(1-p)}$ and I think calculating $\lim_{n\rightarrow \infty}\binom{n}{n_1}$ is the main issue of the problem

edited Nov 02 '14 at 15:31

asked Nov 02 '14 at 14:40

CLAUDE

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Have you tried Stirling's approximation ? Or are you purposely avoiding it ? – Lucian Nov 02 '14 at 16:14
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Stirling definitely works and is probably the best (and easiest) way to prove this. – MHS Nov 03 '14 at 14:11
Related (perhaps dup?) http://math.stackexchange.com/questions/235962/asymptotics-of-binomial-coefficients-and-the-entropy-function – leonbloy Nov 03 '14 at 14:46

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

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