# upper bounds for binomial

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

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

Using binomial theorem and for $x\ge0$:

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

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 $x=\frac{k}{n-k}$

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