For $x\in[0,1]$, define $f_1(x)=1$ and $$ f_{n+1}(x)=(1-x)^n\left(nx+1\right)+f_n(x)\left(1-(1-x)^{n+1}\right)\;\;\mbox{for}\;\;n=1,2,\dots. $$ I want to study some asymptotic results of $f_n$. For instance, to find a function $g(x)$ such that $f_n/g(x)\to1$ as $n\to\infty$. It is seen that $f_n(0)=f_n(1)=1$ for all $n$. Numerical calculations reveals that $f_n$ tends to be stable when $n\ge500$; and the peak occurs very closely to $x=0$. Thus, the point-wise limit function of $f_n$ seems to be discontinuous on $[0,1]$.
Any help of studying the function $g(x)$ will be appreciated.
Attached is the figure of $f_{1010}(x)$ (the blue one) and $1-\log(x)$ (the yellow one) on $[0,1]$.
