Simplify $f(y) = \displaystyle \sum^{\lfloor\frac{1}{y} \rfloor}_{x= 1} xy\left(\frac{1-y}{1-(x-1)y}\right)^{x-1}$, where $0 < y < 0.5$ Kindly simplify $f(y) =  \displaystyle \sum^{\lfloor\frac{1}{y} \rfloor}_{x= 1} xy\left(\frac{1-y}{1-(x-1)y}\right)^{x-1}$, where $0 < y  < 0.5$.

$$f(y) =  \displaystyle \sum^{\lfloor\frac{1}{y} \rfloor}_{x= 1} xy\left(\frac{1-y}{1-(x-1)y}\right)^{x-1},\;\;0<y<1/2.$$
The nontrivial regime is when $y\rightarrow 0$. Then the sum is dominated by the last term, so we can approximate
$$f(y)\approx y\lfloor\frac{1}{y} \rfloor \left(\frac{1-y}{1+y-y\lfloor\frac{1}{y} \rfloor}\right)^{\lfloor\frac{1}{y} \rfloor-1}.$$
This is actually quite accurate already for $y\lesssim 0.2$, see the plot where the exact result (red) is almost indistinguishable from the approximation (green).
The upper envelope of the zigzag curve (orange) is given by the points where $1/y$ is an integer, so by setting $x=1/y$,
$$f_{\rm upper-envelope}(y)=(1/y-1)^{1/y-1}.$$
The lower envelope (blue) is obtained by setting $x=1/y-1$,
$$f_{\rm lower-envelope}(y)=(2y)^{2-1/y} (1-y)^{1/y-1} .$$

