Lets say you have a partial fraction of the form:
$$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$
Can anyone explain to me, in mildly plain English, how to convert this partial fraction to a continued fraction of the form:
$$ f(x)= b_0+\frac{b_1}{1+}\;\;\frac{b_2x}{1+}\;\;\frac{b_3x}{1+}\;\;... $$
I have tried to find good answers but have only been able to find very difficult academic papers detailing QD algorithms (Rutishauser, 1954), Jacobi matrices (de Boor & Golub, 1978), etc. I have also looked in the textbook Numerical Methods That Work (Acton) but he only has a small section on QD algorithms.
Any help or direction to a good resource is greatly appreciated.