monotonicity of a function with integer numbers

Question

How to prove \begin{equation} f(x,N)=\frac{1-(1-\frac{x}{N})^{N}}{x} \end{equation} is a monotonically decreasing function of $x$ with $x \in [0, 1]$ and for any N=1,2,....

I plot the function for several $N$ which shows the function is monotonically decreasing function of $x$. How to prove it theoretically.

Answer 1

Expand $(1-\frac{x}{N})^N = 1-N\frac{x}{N}+\frac{N(N-1)}{2}\frac{x^2}{N^2}-\frac{N(N-1)(N-2)}{3!}\frac{x^3}{N^3}+O(x^4)$. Then we have $f(x,N)=1-\frac{(N-1)}{2}\frac{x}{N}+\frac{(N-1)(N-2)}{3!}\frac{x^2}{N^2}-O(x^3)$ Taking the derivative of $f(x,N)$, we have $f'(x,N)=-\frac{N-1}{2N}+\frac{2x(N-1)(N-2)}{3!N^2}-O(x^2)\leq 0$ for all $x\in [0,1]$.