Lately, I learned about the following continued fraction for the exponential function:
$$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$
I thought it was something new, but evaluating the successive convergents of this continued fraction was a disappointment, as they are nothing more than the partial sums of the usual series $\exp(x)=\sum_{j=0}^{\infty}\frac{x^j}{j!}$.
So, there must be some way to obtain the continued fraction from the series. How might this be done?