Inspired by On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$? I made an exercise for the student who are interested by the problem :
Problem :
1).Determine when occurs the relation :$$C+\sum_{i=1}^{3}\frac{x^{n}}{n!}=numerator(ax(b_{0}\left(ax+1\right)+b_{1}\cdot\frac{1}{ax+1})$$
Numerator means we take only the numerator without coefficient in factor which are not coprime .(see the good observation due to Henrik support the community)
2).Using 1). Build a polynomial $f(x)$ which converges to $e^x$
3).Setting $x=0$ after some (numerical works) of iteration with $f_{iteration}(x)$ obtain using a rational approximation of $e^b$ :
$$y\simeq P$$
Where $P$ is a positive rational number .
4).Evaluate the speed of convergence to $y$ .
I can answer the first question globaly . I'm really curious for the last question .
1).
$$2/3x(9/16(2/3x+1)+((15)/9+16/9*y)*9/(16)/(1+2/3x))-y=3(1/6x^3+1/2x^2+x-y)/((2x+3))$$
Wich is the case $n=3$
2).$$\frac{2}{3}x\left(\frac{9}{16}\left(\frac{2}{3}x+1\right)+\frac{\left(\frac{15}{9}+\frac{16}{9}y\right)\frac{9}{16}}{1+\frac{2}{3}x}\right)-y+\frac{x^{4}}{4\left(4x+6\right)}$$ Question :
Can you give a clear answer for the exercise ?
