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I'm trying to calculate the upper bound of the binomial coefficient:

\begin{equation} \sum\limits_{j=0}^{k} {n\choose j}<\left( \frac{ne}{k} \right)^k \end{equation}

Using binomial theorem and for $x\ge0$:

\begin{equation} \sum\limits_{j=0}^{k} {n\choose j}{x^j}\le(1+x)^n \end{equation}

dividing both sides by $x^k$,we obtain:

$$ \sum\limits_{j=0}^{k} {n\choose j}{\frac{1}{x^{k-j}}}\le\frac{(1+x)^n}{x^k} $$

For x<1 the term $\frac{(1+x)^n}{x^k}$

obtain his minimum value at point $x=\frac{k}{n-k}$

I don't understand why... Could you please help. I thank you in advance.

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up vote 1 down vote accepted

Consider $f(x) = \frac{(1+x)^n}{x^k}$ and apply your favorite calculus-based test for local extrema.

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i got it ! thank you – david Jun 15 '12 at 21:14

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