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.